text: direction with low sidelobe level and scrambled phases in other sensitive adaptive array. IEEE Transactions on Antennas directions. and Propagation, 29(5):718–725, September 1981. REFERENCE [13] A. Nehorai and E. Paldi. Digital Signal Processing Handbook, chapter Electromagnetic vector-sensor array [1] A. Babakhani, D. B. Rutledge, and A. Hajimiri. Trans- processing, pages 65.1–65.26. CRC Press, 1998. mitter architectures based on near-ﬁeld direct antenna [14] S. Miron, N. Le Bihan, and J. I. Mars. Quaternion- modulation. IEEE Journal of Solid-State Circuits, MUSIC for vector-sensor array processing. IEEE Trans- 43(12):2674–2692, December 2008. actions on Signal Processing, 54(4):1218–1229, April [2] A. Babakhani, D. B. Rutledge, and A. Hajimiri. Near- 2006. ﬁeld direct antenna modulation. IEEE Microwave Mag- [15] X. M. Gou, Y. G. Xu, Z. W. Liu, and X. F. Gong. azine, 10(1):36–46, February 2009. Quaternion-Capon beamformer using crossed-dipole ar- [3] M. P. Daly and J. T. Bernhard. Beamsteering in pattern rays. In Proc. IEEE International Symposium on Mi- reconﬁgurable arrays using directional modulation. IEEE crowave, Antenna, Propagation, and EMC Technologies Transactions on Antennas and Propagation, 58(7):2259– for Wireless Communications (MAPE), pages 34–37, 2265, March 2010. Beijing, China, November 2011. [4] M. P. Daly and J. T. Bernhard. Directional modulation [16] W. Liu. Antenna array signal processing for a quaternion- technique for phased arrays. IEEE Transactions on valued wireless communication system. In Proc. the Antennas and Propagation, 57(9):2633–2640, September Benjamin Franklin Symposium on Microwave and An- 2009. tenna Sub-systems (BenMAS), Philadelphia, US, Septem- [5] B. Zhang, W. Liu, and X. Gou. Compressive sensing ber 2014. based sparse antenna array design for directional modula- [17] M. B. Hawes and W. Liu. Design of ﬁxed beamformers tion. IET Microwaves, Antennas Propagation, 11(5):634– based on vector-sensor arrays. International Journal 641, April 2017. of Antennas and Propagation, 2015:181937.1–181937.9, [6] T. Hong, M. Z. Song, and Y. Liu. Dual-beam directional March 2015. modulation technique for physical-layer secure commu- [18] J. J. Xiao and A. Nehorai. Optimal polarized beampattern nication. IEEE Antennas and Wireless Propagation synthesis using a vector antenna array. IEEE Trans- Letters, 10:1417–1420, December 2011. actions on Signal Processing, 57(2):576–587, February [7] H. Z. Shi and A. Tennant. Enhancing the security of 2009. communication via directly modulated antenna arrays. [19] O. M. Isaeva and V. A. Sarytchev. Quaternion presen- IET Microwaves, Antennas & Propagation, 7(8):606– tations polarization state. In Proc. 2nd IEEE Topical 611, June 2013. Symposium of Combined Optical-Microwave Earth and [8] Y. Ding and V. Fusco. Directional modulation transmitter Atmosphere Sensing, pages 195–196, Atlanta, US, April radiation pattern considerations. IET Microwaves, Anten- 1995. nas & Propagation, 7(15):1201–1206, December 2013. [20] L. H. Zetterberg and H. Brandstrom. Codes for com- [9] Y. Ding and V. Fusco. A vector approach for the bined phase and amplitude modulated signals in a four- analysis and synthesis of directional modulation transmit- dimensional space. IEEE Transactions on Communica- ters. IEEE Transactions on Antennas and Propagation, tions, 25(29):943–950, September 1977. 62(1):361–370, January 2014. [21] X. Lan and W. Liu. Fully quaternion-valued adaptive [10] Y. Ding and V. Fusco. Constraining directional mod- beamforming based on crossed-dipole arrays. Electron- ulation transmitter radiation patterns. IET Microwaves, ics, 6(2):34, April 2017. Antennas & Propagation, 8(15):1408–1415, September [22] X. R. Zhang, W. Liu, Y. G. Xu, and Z. W. Liu. 2014. Quaternion-valued robust adaptive beamformer for elec- [11] Q. J. Zhu, S. W. Yang, R. L. Yao, and Z. P. Nie. Di- tromagnetic vector-sensor arrays with worst-case con- rectional modulation based on 4-D antenna arrays. IEEE straint. Signal Processing, 104:274–283, April 2014. Transactions on Antennas and Propagation, 62(2):621– [23] M. B. Hawes, W. Liu, and L. Mihaylova. Compressive 628, February 2014. sensing based design of sparse tripole arrays. Sensors, [12] R. T. Compton. On the performance of a polarization

text: direction

text: with

text: low

text: sidelobe

text: level

text: and

text: scrambled

text: phases

text: in

text: other

text: sensitive

label: reference

text: adaptive

text: array.

text: IEEE

text: Transactions

text: on

text: Antennas

text: directions.

text: and

text: Propagation,

text: September

text: REFERENCE

text: A.

text: Nehorai

text: and

text: E.

text: Paldi.

text: Digital

text: Signal

text: Processing

text: Handbook,

text: chapter

text: Electromagnetic

text: vector-sensor

text: array

text: A.

text: Babakhani,

text: D.

text: B.

text: Rutledge,

text: and

text: A.

text: Hajimiri.

text: Trans-

text: processing,

text: pages

text: CRC

text: Press,

text: mitter

text: architectures

text: based

text: on

text: near-ﬁeld

text: direct

text: antenna

text: S.

text: Miron,

text: N.

text: Le

text: Bihan,

text: and

text: J.

text: I.

text: Mars.

text: Quaternion-

text: modulation.

text: IEEE

text: Journal

text: of

text: Solid-State

text: Circuits,

text: MUSIC

text: for

text: vector-sensor

text: array

text: processing.

text: IEEE

text: Trans-

text: December

text: actions

text: on

text: Signal

text: Processing,

text: April

text: A.

text: Babakhani,

text: D.

text: B.

text: Rutledge,

text: and

text: A.

text: Hajimiri.

text: Near-

text: ﬁeld

text: direct

text: antenna

text: modulation.

text: IEEE

text: Microwave

text: Mag-

text: X.

text: M.

text: Gou,

text: Y.

text: G.

text: Xu,

text: Z.

text: W.

text: Liu,

text: and

text: X.

text: F.

text: Gong.

text: azine,

text: February

text: Quaternion-Capon

text: beamformer

text: using

text: crossed-dipole

text: ar-

text: M.

text: P.

text: Daly

text: and

text: J.

text: T.

text: Bernhard.

text: Beamsteering

text: in

text: pattern

text: rays.

text: In

text: Proc.

text: IEEE

text: International

text: Symposium

text: on

text: Mi-

text: reconﬁgurable

text: arrays

text: using

text: directional

text: modulation.

text: IEEE

text: crowave,

text: Antenna,

text: Propagation,

text: and

text: EMC

text: Technologies

text: Transactions

text: on

text: Antennas

text: and

text: Propagation,

text: for

text: Wireless

text: Communications

text: (MAPE),

text: pages

text: March

text: Beijing,

text: China,

text: November

text: M.

text: P.

text: Daly

text: and

text: J.

text: T.

text: Bernhard.

text: Directional

text: modulation

text: W.

text: Liu.

text: Antenna

text: array

text: signal

text: processing

text: for

text: a

text: quaternion-

text: technique

text: for

text: phased

text: arrays.

text: IEEE

text: Transactions

text: on

text: valued

text: wireless

text: communication

text: system.

text: In

text: Proc.

text: the

text: Antennas

text: and

text: Propagation,

text: September

text: Benjamin

text: Franklin

text: Symposium

text: on

text: Microwave

text: and

text: An-

text: tenna

text: Sub-systems

text: (BenMAS),

text: Philadelphia,

text: US,

text: Septem-

text: B.

text: Zhang,

text: W.

text: Liu,

text: and

text: X.

text: Gou.

text: Compressive

text: sensing

text: ber

text: based

text: sparse

text: antenna

text: array

text: design

text: for

text: directional

text: modula-

text: M.

text: B.

text: Hawes

text: and

text: W.

text: Liu.

text: Design

text: of

text: ﬁxed

text: beamformers

text: tion.

text: IET

text: Microwaves,

text: Antennas

text: Propagation,

text: based

text: on

text: vector-sensor

text: arrays.

text: International

text: Journal

text: April

text: of

text: Antennas

text: and

text: Propagation,

text: T.

text: Hong,

text: M.

text: Z.

text: Song,

text: and

text: Y.

text: Liu.

text: Dual-beam

text: directional

text: March

text: modulation

text: technique

text: for

text: physical-layer

text: secure

text: commu-

text: J.

text: J.

text: Xiao

text: and

text: A.

text: Nehorai.

text: Optimal

text: polarized

text: beampattern

text: nication.

text: IEEE

text: Antennas

text: and

text: Wireless

text: Propagation

text: synthesis

text: using

text: a

text: vector

text: antenna

text: array.

text: IEEE

text: Trans-

text: Letters,

text: December

text: actions

text: on

text: Signal

text: Processing,

text: February

text: H.

text: Z.

text: Shi

text: and

text: A.

text: Tennant.

text: Enhancing

text: the

text: security

text: of

text: communication

text: via

text: directly

text: modulated

text: antenna

text: arrays.

text: O.

text: M.

text: Isaeva

text: and

text: V.

text: A.

text: Sarytchev.

text: Quaternion

text: presen-

text: IET

text: Microwaves,

text: Antennas

text: Propagation,

text: tations

text: polarization

text: state.

text: In

text: Proc.

text: IEEE

text: Topical

text: June

text: Symposium

text: of

text: Combined

text: Optical-Microwave

text: Earth

text: and

text: Y.

text: Ding

text: and

text: V.

text: Fusco.

text: Directional

text: modulation

text: transmitter

text: Atmosphere

text: Sensing,

text: pages

text: Atlanta,

text: US,

text: April

text: radiation

text: pattern

text: considerations.

text: IET

text: Microwaves,

text: Anten-

text: nas

text: Propagation,

text: December

text: L.

text: H.

text: Zetterberg

text: and

text: H.

text: Brandstrom.

text: Codes

text: for

text: com-

text: Y.

text: Ding

text: and

text: V.

text: Fusco.

text: A

text: vector

text: approach

text: for

text: the

text: bined

text: phase

text: and

text: amplitude

text: modulated

text: signals

text: in

text: a

text: four-

text: analysis

text: and

text: synthesis

text: of

text: directional

text: modulation

text: transmit-

text: dimensional

text: space.

text: IEEE

text: Transactions

text: on

text: Communica-

text: ters.

text: IEEE

text: Transactions

text: on

text: Antennas

text: and

text: Propagation,

text: tions,

text: September

text: January

text: X.

text: Lan

text: and

text: W.

text: Liu.

text: Fully

text: quaternion-valued

text: adaptive

text: Y.

text: Ding

text: and

text: V.

text: Fusco.

text: Constraining

text: directional

text: mod-

text: beamforming

text: based

text: on

text: crossed-dipole

text: arrays.

text: Electron-

text: ulation

text: transmitter

text: radiation

text: patterns.

text: IET

text: Microwaves,

text: ics,

text: April

text: Antennas

text: Propagation,

text: September

text: X.

text: R.

text: Zhang,

text: W.

text: Liu,

text: Y.

text: G.

text: Xu,

text: and

text: Z.

text: W.

text: Liu.

text: Quaternion-valued

text: robust

text: adaptive

text: beamformer

text: for

text: elec-

text: Q.

text: J.

text: Zhu,

text: S.

text: W.

text: Yang,

text: R.

text: L.

text: Yao,

text: and

text: Z.

text: P.

text: Nie.

text: Di-

text: tromagnetic

text: vector-sensor

text: arrays

text: with

text: worst-case

text: con-

text: rectional

text: modulation

text: based

text: on

text: antenna

text: arrays.

text: IEEE

text: straint.

text: Signal

text: Processing,

text: April

text: Transactions

text: on

text: Antennas

text: and

text: Propagation,

text: M.

text: B.

text: Hawes,

text: W.

text: Liu,

text: and

text: L.

text: Mihaylova.

text: Compressive

text: February

text: sensing

text: based

text: design

text: of

text: sparse

text: tripole

text: arrays.

text: Sensors,

text: R.

text: T.

text: Compton.

text: On

text: the

text: performance

text: of

text: a

text: polarization


text: 15(12):31056–31068, December 2015.

text: December


text: 12 PIERRE GERMAIN, BENJAMIN HARROP-GRIFFITHS, AND JEREMY L. MARZUOLA (cid:40) We complete this section by considering a priori estimates for a model linear equation, − w + (1 + g)w + βg w + aw + f = νw t yyy y yy y yyyy (2.10) w(0) = 0, ∈ R ≥ where β , ν 0 are constants and g, a, f are suﬃciently smooth functions. This will provide a model for the equation satisﬁed by ∂nW and will subsequently by used to obtain uniform (in ν) y bounds for solutions. Our main a priori estimate for solutions to (2.10) is the following: Proposition 2.7. Let T > 0 and suppose that (cid:107) (cid:107) ≤ 1 (cid:107) (cid:107) (cid:107) (cid:107) (cid:46) (2.11) g ∞ ∞ , g ∞ ∞ + a ∞ ∞ 1. L ([0,T];L ) 2 L ([0,T];W3, ) L ([0,T];W1, ) ≤ (cid:28) Then, if w is a suﬃciently smooth, localized solution to (2.10) and 0 ν 1 is suﬃciently small we have the estimate (cid:107) (cid:107) (cid:107) (cid:107) (cid:46) (cid:107) (cid:107) (2.12) w 2 ∞ + ν w 2 σ(T ) f 2 ∞ , L ([0,T];L2) yy L2((0,T);L2) L ([0,T];L2) where (cid:82) (2.13) σ(T ) = eC 0T(1+(cid:107)gt(cid:107)L∞)dt − 1, and the constants are independent of ν. ≥ Further, we have the weighted estimate for k 1, (cid:32) (cid:33) (cid:107)(cid:104) (cid:105) (cid:107) (cid:107)(cid:104) (cid:105) (cid:107)(cid:88) y kw 2 ∞ + ν y kw 2 L ([0,T];L2) yy L2((0,T);L2) (2.14) (cid:46) (cid:107)(cid:104) (cid:105) (cid:107) 2 (cid:107)(cid:104) (cid:105) − (cid:107) σ(T ) y kf 2 ∞ + y k 1∂nw 2 ∞ , L ([0,T];L2) y L ([0,T];L2) n=0 where again the constants are independent of ν. Proof. Diﬀerentiating with respect to time and integrating by parts we obtain d (cid:107) − (cid:107) (cid:104) − (cid:105) 2 − (cid:104) − (cid:105) dt (1 + g)13β 12 w 2L2 = 2 (1 + g)23β 1w, wt + (3β 1) (1 + g)23β 2gtw, w −1 (cid:104) − (cid:105) (cid:104) − (cid:105) = β ((1 + g)23β 1gy)yyw, w + ((1 + g)23β 1a)yw, w 3 − (cid:104) − (cid:105) 2 − (cid:104) − (cid:105) 2 (1 + g)23β 1f, w + ( β 1) (1 + g)23β 2gtw, w 3 − (cid:107) − (cid:107) (cid:104) − (cid:105) 2ν (1 + g)13β 12 wyy 2L2 + 2ν ((1 + g)23β 1)yywy, wy − (cid:104) − (cid:105) 2ν ((1 + g)23β 1)yyw, wyy . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: PIERRE

text: GERMAIN,

text: BENJAMIN

text: HARROP-GRIFFITHS,

text: AND

text: JEREMY

text: L.

text: MARZUOLA

label: equation

text: We

text: complete

text: this

text: section

text: by

text: considering

text: a

text: priori

text: estimates

text: for

text: a

text: model

text: linear

text: equation,

text: w

text: g)w

text: βg

text: w

text: aw

text: f

text: νw

text: t

text: yyy

text: y

text: yy

text: y

text: yyyy

text: R

text: where

text: β

text: ν

text: are

text: constants

text: and

text: g,

text: a,

text: f

text: are

text: suﬃciently

text: smooth

text: functions.

text: This

text: will

text: provide

text: a

text: model

text: for

text: the

text: equation

text: satisﬁed

text: by

text: ∂nW

text: and

text: will

text: subsequently

text: by

text: used

text: to

text: obtain

text: uniform

text: (in

text: ν)

text: y

text: bounds

text: for

text: solutions.

text: Our

text: main

text: a

text: priori

text: estimate

text: for

text: solutions

text: to

text: is

text: the

text: following:

text: Proposition

text: Let

text: T

text: and

text: suppose

text: that

text: g

text: g

text: a

text: L

text: L

text: L

text: Then,

text: if

text: w

text: is

text: a

text: suﬃciently

text: smooth,

text: localized

text: solution

text: to

text: and

text: ν

text: is

text: suﬃciently

text: small

text: we

text: have

text: the

text: estimate

text: w

text: ν

text: w

text: σ(T

text: f

text: L

text: yy

text: L

text: where

text: σ(T

text: eC

text: and

text: the

text: constants

text: are

text: independent

text: of

text: ν.

text: Further,

text: we

text: have

text: the

text: weighted

text: estimate

text: for

text: k

text: y

text: kw

text: ν

text: y

text: kw

text: L

text: yy

text: σ(T

text: y

text: kf

text: y

text: k

text: L

text: y

text: L

text: where

text: again

text: the

text: constants

text: are

text: independent

text: of

text: ν.

text: Proof.

text: Diﬀerentiating

text: with

text: respect

text: to

text: time

text: and

text: integrating

text: by

text: parts

text: we

text: obtain

text: d

text: dt

text: w

text: wt

text: w

text: −1

text: β

text: w

text: w

text: w

text: β

text: w

text: 2ν

text: wyy

text: 2ν

text: wy

text: 2ν

text: wyy

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A QUASILINEAR KDV EQUATION WITH DEGENERATE DISPERSION 15 H We will then consider the existence of solutions to (3.7) in the weighted Sobolev spaces N,K deﬁned as in (2.1). We now describe our assumptions on the initial data ρ = u2, which are most easily stated in 0 the y-coordinates. However, they may be phrased in the original coordinates using the change of variables (3.4) and we compute a couple of special cases in Section . ≥ We ﬁrst make the assumption that there exists an integer K 0 and some δ > 0 so that in the 0 y-coordinates, (cid:104) (cid:105) ≥ (3.9) in∈Rf ρ(y)56 y K0 δ. y ∈ R It seems reasonable to expect this hypothesis is true whenever (in the x-coordiantes) ρ C3( ) has subcritical decay at both endpoints in the sense of Deﬁnition 1.2. We verify that it is indeed true for polynomially decaying data in Section . Next we assume that (cid:13) (cid:13) (cid:88) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:107) (cid:107) 2K0+7 ∂nρ (cid:46) (3.10) ρ ∞ + y 1. L ρ n=1 L2 M Finally we assume that there exists some > 0 so that the inhomogeneous term F , deﬁned as in (3.6), satisﬁes, (cid:107) (cid:107) ≤ M 5 (3.11) ρ6 F H4,K0 , ≥ where the integer K 0 appears in the lower bound (3.9). 0 Remark 3.1. For most estimates we will treat ρ and its derivatives as coeﬃcients in the linear and nonlinear terms involving W . In this case it will be more convenient to use that from the estimate (3.10) and Sobolev embedding we have the pointw(cid:13)ise b(cid:13)ound, (cid:88) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:107) (cid:107) 2K0+6 ∂nρ (cid:46) (3.12) ρ ∞ + y 1. L ρ ∞ n=1 L The only exception to this will be when 2K + 7 derivatives fall on ρ, where we will instead use 0 the estimate (3.10) directly. ∈ Remark 3.2. We note when µ = 1 the assumptions on ρ do not preclude the case that ρ (Φ )2 + ∞ R − ≤ (cid:28) B,c (cid:15)C ( ) where c > 0, 1c2 B < 0 and 0 < (cid:15) 1, i.e. ρ is a small perturbation of the non- 0 4 B,c degenerate traveling wave Φ . B,c Remark 3.3. The assumptions (3.11),(3.12) are far from the optimal regularity. In future work we will show that it is possible to improve the regularity by taking further advantage of the dispersive smoothing eﬀects similarly to [26,27,41,42]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: QUASILINEAR

text: KDV

text: EQUATION

text: WITH

text: DEGENERATE

text: DISPERSION

text: H

text: We

text: will

text: then

text: consider

text: the

text: existence

text: of

text: solutions

text: to

text: in

text: the

text: weighted

text: Sobolev

text: spaces

text: N,K

text: deﬁned

text: as

text: in

text: We

text: now

text: describe

text: our

text: assumptions

text: on

text: the

text: initial

text: data

text: ρ

text: which

text: are

text: most

text: easily

text: stated

text: in

text: the

text: y-coordinates.

text: However,

text: they

text: may

text: be

text: phrased

text: in

text: the

text: original

text: coordinates

text: using

text: the

text: change

text: of

text: variables

text: and

text: we

text: compute

text: a

text: couple

text: of

text: special

text: cases

text: in

text: Section

text: We

text: ﬁrst

text: make

text: the

text: assumption

text: that

text: there

text: exists

text: an

text: integer

text: K

text: and

text: some

text: δ

text: so

text: that

text: in

text: the

text: y-coordinates,

text: in∈Rf

text: y

text: δ.

text: y

text: R

text: It

text: seems

text: reasonable

text: to

text: expect

text: this

text: hypothesis

text: is

text: true

text: whenever

text: (in

text: the

text: x-coordiantes)

text: ρ

text: has

text: subcritical

text: decay

text: at

text: both

text: endpoints

text: in

text: the

text: sense

text: of

text: Deﬁnition

text: We

text: verify

text: that

text: it

text: is

text: indeed

text: true

text: for

text: polynomially

text: decaying

text: data

text: in

text: Section

text: Next

text: we

text: assume

text: that

text: ∂nρ

text: ρ

text: y

text: L

text: ρ

text: M

text: Finally

text: we

text: assume

text: that

text: there

text: exists

text: some

text: so

text: that

text: the

text: inhomogeneous

text: term

text: F

text: deﬁned

text: as

text: in

text: satisﬁes,

text: M

text: ρ6

text: F

text: where

text: the

text: integer

text: K

text: appears

text: in

text: the

text: lower

text: bound

text: Remark

text: For

text: most

text: estimates

text: we

text: will

text: treat

text: ρ

text: and

text: its

text: derivatives

text: as

text: coeﬃcients

text: in

text: the

text: linear

text: and

text: nonlinear

text: terms

text: involving

text: W

text: In

text: this

text: case

text: it

text: will

text: be

text: more

text: convenient

text: to

text: use

text: that

text: from

text: the

text: estimate

text: and

text: Sobolev

text: embedding

text: we

text: have

text: the

text: ∂nρ

text: ρ

text: y

text: L

text: ρ

text: L

text: The

text: only

text: exception

text: to

text: this

text: will

text: be

text: when

text: derivatives

text: fall

text: on

text: ρ,

text: where

text: we

text: will

text: instead

text: use

text: the

text: estimate

text: directly.

text: Remark

text: We

text: note

text: when

text: µ

text: the

text: assumptions

text: on

text: ρ

text: do

text: not

text: preclude

text: the

text: case

text: that

text: ρ

text: (Φ

text: R

text: B,c

text: where

text: c

text: B

text: and

text: i.e.

text: ρ

text: is

text: a

text: small

text: perturbation

text: of

text: the

text: non-

text: B,c

text: degenerate

text: traveling

text: wave

text: Φ

text: B,c

text: Remark

text: The

text: assumptions

text: are

text: far

text: from

text: the

text: optimal

text: regularity.

text: In

text: future

text: work

text: we

text: will

text: show

text: that

text: it

text: is

text: possible

text: to

text: improve

text: the

text: regularity

text: by

text: taking

text: further

text: advantage

text: of

text: the

text: dispersive

text: smoothing

text: eﬀects

text: similarly

text: to

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A QUASILINEAR KDV EQUATION WITH DEGENERATE DISPERSION 23 (cid:40) ∈ Using this, we ﬁnd the characteristic passing through (t, x) = (0, 0) by ﬁnding a solution ξ C1([0, T ]) of the ODE ξ˙(t) = B(t, 0), (cid:90) ξ(0) = 0. We may then deﬁne x 1 X(t, x) = ξ(t) + ds, 1 + Z(t, s) 0 where we note that from the proof of Theorem 3.4 we have (cid:107) (cid:107) (cid:28) sup Z ∞ 1, ∈ L t [0,T] ∈ × R and hence X C1([0, T ] ). By construction, it satisﬁes ∈ × R X (t, x) = B(t, x) for (t, x) [0, T ] . t (cid:55)→ ∈ × R The map x X(t, x) is a diﬀeomorphism so we may ﬁnd an inverse Y C1([0, T ] ) so that Yx(t, x) = (1 + Z(t, Y (t, x))) and hence ρ(Y (t, x))n3 ∂xn+1Y (t, x) ∈ Cb([0, T ] × R) for n = 1, 2, 3. We then deﬁne u(t, x) = Y (t, x)u (Y (t, x)), x 0 ∈ × R ∈ R and observe that u C1([0, T ] ) and u2 C([0, T ]; C3( )). Further, with this deﬁnition we see b b that B(t, x) = b(t, X(t, x)), where b = 1(u2) + µu2. In particular, X satisﬁes the ODE (1.11) (recalling that Z(0, x) = 0) and 2 xx hence u is a hydrodynamic solution of (1.1). Further, using the bounds on Y it is straightforward to verify that u43 ∈ C([0, T ]; Cb2(R)) and hence satisﬁes the hypothesis of Theorem 1.4. We now prove Theorem 1.4: the uniqueness of hydrodynamic solutions. We ﬁrst note that if u is a hydrodynamic solution of (1.1) then w = u2 is a non-negative classical (cid:40) (cid:0) (cid:1) (cid:0) (cid:1) solution of the equation w + 2 1w + µw w + 1w + µw w = 0, (4.1) t 2 xx x 2 xx x w(0) = w := u2. 0 0 Next we deﬁne the Lagrangian map X as in (1.11) and, treating b as a ﬁxed function, uniqueness of solutions to linear transport equations ensures that u may be written in the form, u(t, x) = Y (t, x)u (Y (t, x)), x 0 ∈ × R (cid:90) (cid:55)→ where Y C1([0, T ] ) is the inverse of the map x X(t, x). From the ODE satisﬁed by Y we x obtain the estimate | | (cid:46) t (cid:107) (cid:107) ln Y u2 ∞ ds, x W3, ≥ ≥ 0 so as u 0 we have u 0. In particular, provided classical solutions to (4.1) are unique, so are 0 hydrodynamic solutions to (1.1). (cid:40) Taking w the ﬁrst variation w in the equation (4.1), we have w + ww + 1w w + (1w + 3µw)w + (w + 3µw )w = 0, (4.2) t xxx 2 x xx 2 xx x xxx x w(0) = w . 0 We then have the following lemma: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: QUASILINEAR

text: KDV

text: EQUATION

text: WITH

text: DEGENERATE

text: DISPERSION

text: Using

text: this,

text: we

text: ﬁnd

text: the

text: characteristic

text: passing

text: through

text: (t,

text: x)

text: by

text: ﬁnding

text: a

text: solution

text: ξ

text: T

text: of

text: the

text: ODE

text: ξ˙(t)

text: B(t,

text: We

text: may

text: then

text: deﬁne

text: x

text: X(t,

text: x)

text: ξ(t)

text: ds,

text: Z(t,

text: s)

text: where

text: we

text: note

text: that

text: from

text: the

text: proof

text: of

text: Theorem

text: we

text: have

text: sup

text: Z

text: L

text: t

text: R

text: and

text: hence

text: X

text: T

text: By

text: construction,

text: it

text: satisﬁes

text: R

text: X

text: (t,

text: x)

text: B(t,

text: x)

text: for

text: (t,

text: x)

text: T

text: t

text: R

text: The

text: map

text: x

text: X(t,

text: x)

text: is

text: a

text: diﬀeomorphism

text: so

text: we

text: may

text: ﬁnd

text: an

text: inverse

text: Y

text: T

text: so

text: that

text: Yx(t,

text: x)

text: Z(t,

text: Y

text: (t,

text: x)))

text: and

text: hence

text: ρ(Y

text: (t,

text: (t,

text: x)

text: T

text: R)

text: for

text: n

text: We

text: then

text: deﬁne

text: u(t,

text: x)

text: Y

text: (t,

text: x)u

text: (Y

text: (t,

text: x)),

text: x

text: R

text: R

text: and

text: observe

text: that

text: u

text: T

text: and

text: T

text: Further,

text: with

text: this

text: deﬁnition

text: we

text: see

text: b

text: b

text: that

text: B(t,

text: x)

text: b(t,

text: X(t,

text: x)),

text: where

text: b

text: In

text: particular,

text: X

text: satisﬁes

text: the

text: ODE

text: (recalling

text: that

text: x)

text: and

text: xx

text: hence

text: u

text: is

text: a

text: hydrodynamic

text: solution

text: of

text: Further,

text: using

text: the

text: bounds

text: on

text: Y

text: it

text: is

text: straightforward

text: to

text: verify

text: that

text: T

text: and

text: hence

text: satisﬁes

text: the

text: hypothesis

text: of

text: Theorem

text: We

text: now

text: prove

text: Theorem

text: the

text: uniqueness

text: of

text: hydrodynamic

text: solutions.

text: We

text: ﬁrst

text: note

text: that

text: if

text: u

text: is

text: a

text: hydrodynamic

text: solution

text: of

text: then

text: w

text: is

text: a

text: non-negative

text: classical

text: solution

text: of

text: the

text: equation

text: w

text: µw

text: w

text: µw

text: w

text: t

text: xx

text: x

text: xx

text: x

text: w

text: Next

text: we

text: deﬁne

text: the

text: Lagrangian

text: map

text: X

text: as

text: in

text: and,

text: treating

text: b

text: as

text: a

text: ﬁxed

text: function,

text: uniqueness

text: of

text: solutions

text: to

text: linear

text: transport

text: equations

text: ensures

text: that

text: u

text: may

text: be

text: written

text: in

text: the

text: form,

text: u(t,

text: x)

text: Y

text: (t,

text: x)u

text: (Y

text: (t,

text: x)),

text: x

text: R

text: where

text: Y

text: T

text: is

text: the

text: inverse

text: of

text: the

text: map

text: x

text: X(t,

text: x).

text: From

text: the

text: ODE

text: satisﬁed

text: by

text: Y

text: we

text: x

text: obtain

text: the

text: estimate

text: t

text: ln

text: Y

text: ds,

text: x

text: so

text: as

text: u

text: we

text: have

text: u

text: In

text: particular,

text: provided

text: classical

text: solutions

text: to

text: are

text: unique,

text: so

text: are

text: hydrodynamic

text: solutions

text: to

text: Taking

text: w

text: the

text: ﬁrst

text: variation

text: w

text: in

text: the

text: equation

text: we

text: have

text: w

text: ww

text: w

text: (w

text: )w

text: t

text: xxx

text: x

text: xx

text: xx

text: x

text: xxx

text: x

text: w

text: We

text: then

text: have

text: the

text: following

text: lemma:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A QUASILINEAR KDV EQUATION WITH DEGENERATE DISPERSION 29 [22] P. Germain, B. Harrop-Griﬃths, and J. L. Marzuola. Compactons and their variational properties for degenerate KdV and NLS in dimension 1. ArXiv e-prints, Sept. 2017. [23] L. Giacomelli, M. V. Gnann, H. Knu¨pfer, and F. Otto. Well-posedness for the Navier-slip thin-ﬁlm equation in the case of complete wetting. J. Diﬀerential Equations, 257(1):15–81, 2014. [24] G. Glaeser. Racine carr´ee d’une fonction diﬀ´erentiable. Ann. Inst. Fourier (Grenoble), 13(fasc. 2):203–210, 1963. [25] M. V. Gnann. On the regularity for the Navier-slip thin-ﬁlm equation in the perfect wetting regime. Arch. Ration. Mech. Anal., 222(3):1285–1337, 2016. [26] B. Harrop-Griﬃths. Large data local well-posedness for a class of KdV-type equations. Trans. Amer. Math. Soc., 367(2):755–773, 2015. [27] B. Harrop-Griﬃths. Large data local well-posedness for a class of KdV-type equations II. Int. Math. Res. Not. IMRN, (18):8590–8619, 2015. [28] J. K. Hunter. Nonlinear hyperbolic surface waves. In Nonlinear conservation laws and applications, volume 153 of IMA Vol. Math. Appl., pages 303–314. Springer, New York, 2011. [29] J. K. Hunter and R. Saxton. Dynamics of director ﬁelds. SIAM J. Appl. Math., 51(6):1498–1521, 1991. [30] J. Jang and N. Masmoudi. Well-posedness for compressible Euler equations with physical vacuum singularity. Comm. Pure Appl. Math., 62(10):1327–1385, 2009. [31] C. E. Kenig, G. Ponce, and L. Vega. Higher-order nonlinear dispersive equations. Proc. Amer. Math. Soc., 122(1):157–166, 1994. [32] C. E. Kenig, G. Ponce, and L. Vega. On the hierarchy of the generalized KdV equations. In Singular limits of dispersive waves (Lyon, 1991), volume 320 of NATO Adv. Sci. Inst. Ser. B Phys., pages 347–356. Plenum, New York, 1994. [33] C. E. Kenig, G. Ponce, and L. Vega. The Cauchy problem for quasi-linear Schr¨odinger equations. Invent. Math., 158(2):343–388, 2004. [34] C. E. Kenig and G. Staﬃlani. Local well-posedness for higher order nonlinear dispersive systems. J. Fourier Anal. Appl., 3(4):417–433, 1997. [35] H. Knu¨pfer. Well-posedness for the Navier slip thin-ﬁlm equation in the case of partial wetting. Comm. Pure Appl. Math., 64(9):1263–1296, 2011. [36] H. Knu¨pfer. Well-posedness for a class of thin-ﬁlm equations with general mobility in the regime of partial wetting. Arch. Ration. Mech. Anal., 218(2):1083–1130, 2015. [37] H. Knu¨pfer and N. Masmoudi. Darcy’s ﬂow with prescribed contact angle: well-posedness and lubrication ap- proximation. Arch. Ration. Mech. Anal., 218(2):589–646, 2015. [38] H. Koch. Non-Euclidean Singular Integrals and the Porous Medium Equation. Habilitation Thesis. Ruprecht- Karls-Universita¨t Heidelberg, 1999. [39] D. Lannes and G. Metivier. The shoreline problem for the one-dimensional shallow water and Green-Naghdi equations. ArXiv e-prints, Oct. 2017. ∞ [40] R. Manfrin. Well posedness in the C class for u = a(u)∆u. Nonlinear Anal., 36(2, Ser. A: Theory tt Methods):177–212, 1999. [41] J. L. Marzuola, J. Metcalfe, and D. Tataru. Quasilinear Schro¨dinger equations I: Small data and quadratic interactions. Adv. Math., 231(2):1151–1172, 2012. [42] J. L. Marzuola, J. Metcalfe, and D. Tataru. Quasilinear Schro¨dinger equations, II: Small data and cubic nonlin- earities. Kyoto J. Math., 54(3):529–546, 2014. [43] S. Mizohata. On the Cauchy problem, volume 3 of Notes and Reports in Mathematics in Science and Engineering. Academic Press, Inc., Orlando, FL; Science Press Beijing, Beijing, 1985. [44] V. Nesterenko. Dynamics of Heterogeneous Materials. Springer, New York, 2001. [45] D. Pilod. On the Cauchy problem for higher-order nonlinear dispersive equations. J. Diﬀerential Equations, 245(8):2055–2077, 2008. [46] M. A. Porter, C. Daraio, I. Szelengowicz, E. B. Herbold, and P. Kevrekidis. Highly nonlinear solitary waves in heterogeneous periodic granular media. Physica D: Nonlinear Phenomena, 238(6):666 – 676, 2009. [47] P. Rosenau. Nonlinear dispersion and compact structures. Phys. Rev. Lett., 73(13):1737–1741, 1994. [48] P. Rosenau. What is. . . a compacton? Notices Amer. Math. Soc., 52(7):738–739, 2005. [49] P. Rosenau. On a model equation of traveling and stationary compactons. Physics Letters A, 356(1):44 – 50, 2006. [50] P. Rosenau. Compact breathers in a quasi-linear Klein-Gordon equation. Phys. Lett. A, 374(15-16):1663–1667, 2010. [51] P. Rosenau and J. M. Hyman. Compactons: solitons with ﬁnite wavelength. Physical Review Letters, 70(5):564, 1993. [52] P. Rosenau and S. Schochet. Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit. Chaos, 15(1):015111, 18, 2005.

text: A

text: QUASILINEAR

text: KDV

text: EQUATION

text: WITH

text: DEGENERATE

text: DISPERSION

text: P.

text: Germain,

text: B.

text: Harrop-Griﬃths,

text: and

text: J.

text: L.

text: Marzuola.

text: Compactons

text: and

text: their

text: variational

text: properties

text: for

text: degenerate

text: KdV

text: and

text: NLS

text: in

text: dimension

text: ArXiv

text: e-prints,

text: Sept.

text: L.

text: Giacomelli,

text: M.

text: V.

text: Gnann,

text: H.

text: Knu¨pfer,

text: and

text: F.

text: Otto.

text: Well-posedness

text: for

text: the

text: Navier-slip

text: thin-ﬁlm

text: equation

text: in

text: the

text: case

text: of

text: complete

text: wetting.

text: J.

text: Diﬀerential

text: Equations,

text: G.

text: Glaeser.

text: Racine

text: carr´ee

text: d’une

text: fonction

text: diﬀ´erentiable.

text: Ann.

text: Inst.

text: Fourier

text: (Grenoble),

text: M.

text: V.

text: Gnann.

text: On

text: the

text: regularity

text: for

text: the

text: Navier-slip

text: thin-ﬁlm

text: equation

text: in

text: the

text: perfect

text: wetting

text: regime.

text: Arch.

text: Ration.

text: Mech.

text: Anal.,

text: B.

text: Harrop-Griﬃths.

text: Large

text: data

text: local

text: well-posedness

text: for

text: a

text: class

text: of

text: KdV-type

text: equations.

text: Trans.

text: Amer.

text: Math.

text: Soc.,

text: B.

text: Harrop-Griﬃths.

text: Large

text: data

text: local

text: well-posedness

text: for

text: a

text: class

text: of

text: KdV-type

text: equations

text: II.

text: Int.

text: Math.

text: Res.

text: Not.

text: IMRN,

text: J.

text: K.

text: Hunter.

text: Nonlinear

text: hyperbolic

text: surface

text: waves.

text: In

text: Nonlinear

text: conservation

text: laws

text: and

text: applications,

text: volume

text: of

text: IMA

text: Vol.

text: Math.

text: Appl.,

text: pages

text: Springer,

text: New

text: York,

text: J.

text: K.

text: Hunter

text: and

text: R.

text: Saxton.

text: Dynamics

text: of

text: director

text: ﬁelds.

text: SIAM

text: J.

text: Appl.

text: Math.,

text: J.

text: Jang

text: and

text: N.

text: Masmoudi.

text: Well-posedness

text: for

text: compressible

text: Euler

text: equations

text: with

text: physical

text: vacuum

text: singularity.

text: Comm.

text: Pure

text: Appl.

text: Math.,

text: C.

text: E.

text: Kenig,

text: G.

text: Ponce,

text: and

text: L.

text: Vega.

text: Higher-order

text: nonlinear

text: dispersive

text: equations.

text: Proc.

text: Amer.

text: Math.

text: Soc.,

text: C.

text: E.

text: Kenig,

text: G.

text: Ponce,

text: and

text: L.

text: Vega.

text: On

text: the

text: hierarchy

text: of

text: the

text: generalized

text: KdV

text: equations.

text: In

text: Singular

text: limits

text: of

text: dispersive

text: waves

text: (Lyon,

text: volume

text: of

text: NATO

text: Adv.

text: Sci.

text: Inst.

text: Ser.

text: B

text: Phys.,

text: pages

text: Plenum,

text: New

text: York,

text: C.

text: E.

text: Kenig,

text: G.

text: Ponce,

text: and

text: L.

text: Vega.

text: The

text: Cauchy

text: problem

text: for

text: quasi-linear

text: Schr¨odinger

text: equations.

text: Invent.

text: Math.,

text: C.

text: E.

text: Kenig

text: and

text: G.

text: Staﬃlani.

text: Local

text: well-posedness

text: for

text: higher

text: order

text: nonlinear

text: dispersive

text: systems.

text: J.

text: Fourier

text: Anal.

text: Appl.,

text: H.

text: Knu¨pfer.

text: Well-posedness

text: for

text: the

text: Navier

text: slip

text: thin-ﬁlm

text: equation

text: in

text: the

text: case

text: of

text: partial

text: wetting.

text: Comm.

text: Pure

text: Appl.

text: Math.,

text: H.

text: Knu¨pfer.

text: Well-posedness

text: for

text: a

text: class

text: of

text: thin-ﬁlm

text: equations

text: with

text: general

text: mobility

text: in

text: the

text: regime

text: of

text: partial

text: wetting.

text: Arch.

text: Ration.

text: Mech.

text: Anal.,

text: H.

text: Knu¨pfer

text: and

text: N.

text: Masmoudi.

text: Darcy’s

text: ﬂow

text: with

text: prescribed

text: contact

text: angle:

text: well-posedness

text: and

text: lubrication

text: ap-

text: proximation.

text: Arch.

text: Ration.

text: Mech.

text: Anal.,

text: H.

text: Koch.

text: Non-Euclidean

text: Singular

text: Integrals

text: and

text: the

text: Porous

text: Medium

text: Equation.

text: Habilitation

text: Thesis.

text: Ruprecht-

text: Karls-Universita¨t

text: Heidelberg,

text: D.

text: Lannes

text: and

text: G.

text: Metivier.

text: The

text: shoreline

text: problem

text: for

text: the

text: one-dimensional

text: shallow

text: water

text: and

text: Green-Naghdi

text: equations.

text: ArXiv

text: e-prints,

text: Oct.

text: R.

text: Manfrin.

text: Well

text: posedness

text: in

text: the

text: C

text: class

text: for

text: u

text: a(u)∆u.

text: Nonlinear

text: Anal.,

text: Ser.

text: A:

text: Theory

text: tt

text: J.

text: L.

text: Marzuola,

text: J.

text: Metcalfe,

text: and

text: D.

text: Tataru.

text: Quasilinear

text: Schro¨dinger

text: equations

text: I:

text: Small

text: data

text: and

text: quadratic

text: interactions.

text: Adv.

text: Math.,

text: J.

text: L.

text: Marzuola,

text: J.

text: Metcalfe,

text: and

text: D.

text: Tataru.

text: Quasilinear

text: Schro¨dinger

text: equations,

text: II:

text: Small

text: data

text: and

text: cubic

text: nonlin-

text: earities.

text: Kyoto

text: J.

text: Math.,

text: S.

text: Mizohata.

text: On

text: the

text: Cauchy

text: problem,

text: volume

text: of

text: Notes

text: and

text: Reports

text: in

text: Mathematics

text: in

text: Science

text: and

text: Engineering.

text: Academic

text: Press,

text: Inc.,

text: Orlando,

text: FL;

text: Science

text: Press

text: Beijing,

text: Beijing,

text: V.

text: Nesterenko.

text: Dynamics

text: of

text: Heterogeneous

text: Materials.

text: Springer,

text: New

text: York,

text: D.

text: Pilod.

text: On

text: the

text: Cauchy

text: problem

text: for

text: higher-order

text: nonlinear

text: dispersive

text: equations.

text: J.

text: Diﬀerential

text: Equations,

text: M.

text: A.

text: Porter,

text: C.

text: Daraio,

text: I.

text: Szelengowicz,

text: E.

text: B.

text: Herbold,

text: and

text: P.

text: Kevrekidis.

text: Highly

text: nonlinear

text: solitary

text: waves

text: in

text: heterogeneous

text: periodic

text: granular

text: media.

text: Physica

text: D:

text: Nonlinear

text: Phenomena,

text: P.

text: Rosenau.

text: Nonlinear

text: dispersion

text: and

text: compact

text: structures.

text: Phys.

text: Rev.

text: Lett.,

text: P.

text: Rosenau.

text: What

text: is.

text: a

text: compacton?

text: Notices

text: Amer.

text: Math.

text: Soc.,

text: P.

text: Rosenau.

text: On

text: a

text: model

text: equation

text: of

text: traveling

text: and

text: stationary

text: compactons.

text: Physics

text: Letters

text: A,

text: P.

text: Rosenau.

text: Compact

text: breathers

text: in

text: a

text: quasi-linear

text: Klein-Gordon

text: equation.

text: Phys.

text: Lett.

text: A,

text: P.

text: Rosenau

text: and

text: J.

text: M.

text: Hyman.

text: Compactons:

text: solitons

text: with

text: ﬁnite

text: wavelength.

text: Physical

text: Review

text: Letters,

text: P.

text: Rosenau

text: and

text: S.

text: Schochet.

text: Compact

text: and

text: almost

text: compact

text: breathers:

text: a

text: bridge

text: between

text: an

text: anharmonic

text: lattice

text: and

text: its

text: continuum

text: limit.

text: Chaos,


text: 10 PIERRE GERMAIN, BENJAMIN HARROP-GRIFFITHS, AND JEREMY L. MARZUOLA Remark 2.4. In our application of the pointwise estimate (2.8) to control products of functions in H ≤ ≤ − N,K we will require a slight reﬁnement when N is even and n is odd. Suppose that 0 k K 1, ≤ ≤ − (cid:90) (cid:90) ∈ ∞ R that 1 n 2(K k) + N, and that N is even while n is odd. Then, for any f C ( ) we may 0 integrate by parts to obtain (cid:107)(cid:104) (cid:105) (cid:107) − (cid:104) (cid:105) · − − (cid:104) (cid:105) − · − y k+12 ∂ynf 2L2 = y 2k+1∂yn+1f ∂yn 1f dy (2k + 1) y y 2k 1∂ynf ∂yn 1f dy ≤ (cid:107)(cid:104) (cid:105) − (cid:107) (cid:107)(cid:104) (cid:105) (cid:107) (cid:107)(cid:104) (cid:105) − (cid:107) (cid:107)(cid:104) (cid:105) − (cid:107) y k+1∂n 1f y k∂n+1f + y k+1∂n 1f y k 1∂nf y L2 y L2 y L2 y L2 (cid:46) (cid:107) (cid:107) f 2H , N,K − ≤ − where we have used the fact that as n is odd while N is even we have n 1 2(K (k + 1)) + N ≤ − and n + 1 2(K k) + N. Applying the usual 1d Sobolev estimate (2.7) then yields the slight reﬁnement of the pointwise estimate (2.8), (cid:107)(cid:104) (cid:105) − (cid:107) (cid:46) (cid:107) (cid:107) y k+12 ∂yn 1f L∞ f HN,K. (cid:32) (cid:33) (cid:88) (cid:89) Given suﬃciently smooth functions f , . . . , f we deﬁne the multilinear operator 1 M M − − (cid:104) (cid:105) − L [y, f , . . . , f ] = C (y)δ (M 1) y K(M 1) ∂αmf (y) , n 1 M α y m | |≤ α n+≤3 m=1 maxαm n ≥ where δ > 0, K 0 and we assume that the coeﬃcients C are smooth, uniformly bounded α functions. This type of multilinear expression will appear in the perturbative terms of the equation for W considered in Section . We will also use this as a notation, writing G = L [y, f , . . . , f ], n 1 M if a multilinear operator G may be written in this form. We then have the following estimate for multilinear operators of this form: ≤ ≤ ≤ ≤ − ∈ ∞ R Lemma 2.5. Let 0 k K and 0 n 2(K k) + 4. Then if f , . . . , f C ( ) and ∈ ∞ R (cid:32) (cid:33) (cid:32) 1 M(cid:33) 0 C C ( ) we have the estimate α (cid:88) (cid:89) M (cid:107)(cid:104) (cid:105) (cid:107) (cid:46) − − (cid:107) (cid:107) (cid:107) (cid:107) (2.9) y kL [y, f , . . . , f ] δ (M 1) C ∞ f H . n 1 M L2 α L m 4,K α m=1 ≤ ≤ · · · ≤ Proof. First reduction. After reordering the indices, we can assume that α α α . 1 2 ≤ M Let us see quickly why the result is easy if the α are suﬃciently small: ﬁrst, if α 3, the desired ≤ i ≤ M ≤ ≤ − bound easily holds, since k K. Second, assume that α . . . α are all 2, with 1 m K 1. Then one can estimate (cid:104)y(cid:105)K∂yαm(cid:88)fm in(cid:32)L∞, for m ≤ m0,1and mma0tters reduc(cid:89)e to proving t0he(cid:33)desired result for M (cid:48) − − − (cid:104) (cid:105) − − L [y, f , . . . , f ] = C (y)δ (M m0 1) y K(M m0 1) ∂αmf (y) . n m0+1 M α y m | |≤ α n+≤3 m=m0+1 maxαm n ≥ ≤ − ≥ In other words, we can assume in the following that α 3 for m M 1 and α 4. m M ##LTLine## ##LTLine## ##LTLine##

text: PIERRE

text: GERMAIN,

text: BENJAMIN

text: HARROP-GRIFFITHS,

text: AND

text: JEREMY

text: L.

text: MARZUOLA

text: Remark

text: In

text: our

text: application

text: of

text: the

text: pointwise

text: estimate

text: to

text: control

text: products

text: of

text: functions

text: in

text: H

text: N,K

text: we

text: will

text: require

text: a

text: slight

text: reﬁnement

text: when

text: N

text: is

text: even

text: and

text: n

text: is

text: odd.

text: Suppose

text: that

text: k

text: K

text: R

text: that

text: n

text: k)

text: N,

text: and

text: that

text: N

text: is

text: even

text: while

text: n

text: is

text: odd.

text: Then,

text: for

text: any

text: f

text: C

text: we

text: may

text: integrate

text: by

text: parts

text: to

text: obtain

text: y

text: ∂ynf

text: y

text: ∂yn

text: dy

text: y

text: y

text: ∂yn

text: dy

text: y

text: y

text: y

text: y

text: k

text: y

text: y

text: y

text: y

text: f

text: N,K

text: where

text: we

text: have

text: used

text: the

text: fact

text: that

text: as

text: n

text: is

text: odd

text: while

text: N

text: is

text: even

text: we

text: have

text: n

text: (k

text: N

text: and

text: n

text: k)

text: N.

text: Applying

text: the

text: usual

text: Sobolev

text: estimate

text: then

text: yields

text: the

text: slight

text: reﬁnement

text: of

text: the

text: pointwise

text: estimate

text: y

text: ∂yn

text: L∞

text: f

text: HN,K.

text: Given

text: suﬃciently

text: smooth

text: functions

text: f

text: f

text: we

text: deﬁne

text: the

text: multilinear

text: operator

text: M

text: M

text: L

text: [y,

text: f

text: f

text: C

text: (y)δ

text: (M

text: y

text: K(M

text: ∂αmf

text: (y)

text: n

text: M

text: α

text: y

text: m

text: α

text: maxαm

text: n

text: where

text: δ

text: K

text: and

text: we

text: assume

text: that

text: the

text: coeﬃcients

text: C

text: are

text: smooth,

text: uniformly

text: bounded

text: α

text: functions.

text: This

text: type

text: of

text: multilinear

text: expression

text: will

text: appear

text: in

text: the

text: perturbative

text: terms

text: of

text: the

text: equation

text: for

text: W

text: considered

text: in

text: Section

text: We

text: will

text: also

text: use

text: this

text: as

text: a

text: notation,

text: writing

text: G

text: L

text: [y,

text: f

text: f

text: n

text: M

text: if

text: a

text: multilinear

text: operator

text: G

text: may

text: be

text: written

text: in

text: this

text: form.

text: We

text: then

text: have

text: the

text: following

text: estimate

text: for

text: multilinear

text: operators

text: of

text: this

text: form:

text: R

text: Lemma

text: Let

text: k

text: K

text: and

text: n

text: k)

text: Then

text: if

text: f

text: f

text: C

text: and

text: R

text: C

text: C

text: we

text: have

text: the

text: estimate

text: α

text: M

text: y

text: kL

text: [y,

text: f

text: f

text: δ

text: (M

text: C

text: f

text: H

text: n

text: M

text: α

text: L

text: m

text: α

text: Proof.

text: First

text: reduction.

text: After

text: reordering

text: the

text: indices,

text: we

text: can

text: assume

text: that

text: α

text: α

text: α

text: M

text: Let

text: us

text: see

text: quickly

text: why

text: the

text: result

text: is

text: easy

text: if

text: the

text: α

text: are

text: suﬃciently

text: small:

text: ﬁrst,

text: if

text: α

text: the

text: desired

text: i

text: M

text: bound

text: easily

text: holds,

text: since

text: k

text: K.

text: Second,

text: assume

text: that

text: α

text: α

text: are

text: all

text: with

text: m

text: K

text: Then

text: one

text: can

text: estimate

text: for

text: m

text: to

text: proving

text: result

text: for

text: M

text: L

text: [y,

text: f

text: f

text: C

text: (y)δ

text: (M

text: y

text: K(M

text: ∂αmf

text: (y)

text: n

text: M

text: α

text: y

text: m

text: α

text: maxαm

text: n

text: In

text: other

text: words,

text: we

text: can

text: assume

text: in

text: the

text: following

text: that

text: α

text: for

text: m

text: M

text: and

text: α

text: m

text: M

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: BANACH-COLMEZ ET FAISCEAUX COHÉRENTS SUR LA COURBE DE FARGUES-FONTAINE 13 Proposition 3.6. — Soit I un ensemble ordonné ﬁltrant ayant une partie coﬁnale dénom- F brable et ( ) ∈ un système projectif de faisceaux abéliens pour la topologie pro-étale sur un i i I espace adique analytique X, avec morphismes de transition surjectifs. Alors ∀ F k > 0, Rk l←im− = 0. i i Démonstration. — Voir [8, Prop. 3.1.10]. ‹ ‹ On utilisera également le résultat de comparaison suivant. F Proposition 3.7. — Soit X un espace adique analytique sur C et un faisceau étale de → groupes abéliens sur X. Notons ν : X X le morphisme de topos. Alors pour tout ≥ pro´et ´et i 0, ∗F F Hi (X, ν ) = Hi (X, ). pro´et ´et (cid:5) Démonstration. — Ceci est démontré dans [52, Prop. 14.8], avec X remplacé par X , le diamant associé à l’espace adique analytique X. Or la cohomologie pro-étale de X est la (cid:5) (cid:39) même que celle de X , par déﬁnition des sites pro-étales et l’équivalence Perf (cid:5) C,pro´et Perf . La cohomologie étale de X est aussi la même que celle de X car les sites étales C(cid:91),pro´et(cid:5) de X et X sont les mêmes : en eﬀet, les morphismes étales descendent pour la topologie pro-étale (17) (voir la deuxième partie de la preuve de [52, Prop. 9.7]). Proposition 3.8. — On note D le disque unité ouvert de dimension 1 sur C. On a O Hi(D, Q ) = 0 pour tout i > 1. De plus, H1(D, Q ) s’identiﬁe à l’espace (D) des p p 0 fonctions rigides analytiques sur D nulles en zéro. (cid:104) − (cid:105) Démonstration. — Pour tout n > 1, notons B¯ = Spa(C p 1/nT ) la boule fermée de rayon − n p 1/n. Montrons dans un premier temps que Hi (B¯ , Q ) (18) est nul pour i > 1. D’après [3, Th. ´et n p 4.2.6], Hi (B¯ , Z/pk) = 0 si i > 2. On a donc Hi (B¯ , Q ) = 0 si i > 2. La suite exacte de ´et n ´et n p Kummer donne une suite exacte longue → O ∗ O ∗ → → 0 (B¯ ) /( (B¯ ) )pk H1 (B¯ , Z/pk) H1 (B¯ , G ) n n ´et n ´et n m → → → H1 (B¯ , G ) H2 (B¯ , Z/pk) H2 (B¯ , G ). ´et n m ´et n ´et n m Or d’après [3, Lem. 6.1.2], H2 (B¯ , G ) = 0 ; comme on a aussi Pic(B¯ ) = 0 ([43, Satz 1]), ´et n m n on obtient H2 (B¯ , Z/pk) = 0. ´et n On en déduit que H2 (B¯ , Q ) = 0. ´et n p Calculons maintenant H1(D, Q ). On regarde la suite exacte (19) : p → → → → 0 R1l←im− H0(B¯ , Q ) H1(D, Q ) l←im− H1(B¯ , Q ) 0. n p p n p n n 17. Et même pour la v-topologie, cf. [52, Prop 9.7]. 18. Si X est un espace adique, pour nous H´eit(X,Zp) = ←lim−H´eit(X,Z/pk) et H´eit(X,Qp) = H´eit(X,Zp)[1/p], par déﬁnition. 19. Dont l’existence découle de la proposition 3.6, mais se vériﬁe facilement dans ce cas en utilisant [17, Prop. 13.3.1]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BANACH-COLMEZ

text: ET

text: FAISCEAUX

text: COHÉRENTS

text: SUR

text: LA

text: COURBE

text: DE

text: FARGUES-FONTAINE

text: Proposition

text: Soit

text: I

text: un

text: ensemble

text: ordonné

text: ﬁltrant

text: ayant

text: une

text: partie

text: coﬁnale

text: dénom-

text: F

text: brable

text: et

text: un

text: système

text: projectif

text: de

text: faisceaux

text: abéliens

text: pour

text: la

text: topologie

text: pro-étale

text: sur

text: un

text: i

text: i

text: I

text: espace

text: adique

text: analytique

text: X,

text: avec

text: morphismes

text: de

text: transition

text: surjectifs.

text: Alors

text: F

text: k

text: Rk

text: l←im−

text: i

text: i

text: Démonstration.

text: Voir

text: Prop.

text: On

text: utilisera

text: également

text: le

text: résultat

text: de

text: comparaison

text: suivant.

text: F

text: Proposition

text: Soit

text: X

text: un

text: espace

text: adique

text: analytique

text: sur

text: C

text: et

text: un

text: faisceau

text: étale

text: de

text: groupes

text: abéliens

text: sur

text: X.

text: Notons

text: ν

text: X

text: X

text: le

text: morphisme

text: de

text: topos.

text: Alors

text: pour

text: tout

text: pro´et

text: ´et

text: i

text: ∗F

text: F

text: Hi

text: (X,

text: ν

text: Hi

text: (X,

text: pro´et

text: ´et

text: Démonstration.

text: Ceci

text: est

text: démontré

text: dans

text: Prop.

text: avec

text: X

text: remplacé

text: par

text: X

text: le

text: diamant

text: associé

text: à

text: l’espace

text: adique

text: analytique

text: X.

text: Or

text: la

text: cohomologie

text: pro-étale

text: de

text: X

text: est

text: la

text: même

text: que

text: celle

text: de

text: X

text: par

text: déﬁnition

text: des

text: sites

text: pro-étales

text: et

text: l’équivalence

text: Perf

text: C,pro´et

text: Perf

text: La

text: cohomologie

text: étale

text: de

text: X

text: est

text: aussi

text: la

text: même

text: que

text: celle

text: de

text: X

text: car

text: les

text: sites

text: étales

text: de

text: X

text: et

text: X

text: sont

text: les

text: mêmes

text: en

text: eﬀet,

text: les

text: morphismes

text: étales

text: descendent

text: pour

text: la

text: topologie

text: pro-étale

text: (voir

text: la

text: deuxième

text: partie

text: de

text: la

text: preuve

text: de

text: Prop.

text: Proposition

text: On

text: note

text: D

text: le

text: disque

text: unité

text: ouvert

text: de

text: dimension

text: sur

text: C.

text: On

text: a

text: O

text: Hi(D,

text: Q

text: pour

text: tout

text: i

text: De

text: plus,

text: Q

text: s’identiﬁe

text: à

text: l’espace

text: (D)

text: des

text: p

text: p

text: fonctions

text: rigides

text: analytiques

text: sur

text: D

text: nulles

text: en

text: zéro.

text: Démonstration.

text: Pour

text: tout

text: n

text: notons

text: B¯

text: Spa(C

text: p

text: la

text: boule

text: fermée

text: de

text: rayon

text: n

text: p

text: Montrons

text: dans

text: un

text: premier

text: temps

text: que

text: Hi

text: (B¯

text: Q

text: est

text: nul

text: pour

text: i

text: D’après

text: Th.

text: ´et

text: n

text: p

text: Hi

text: (B¯

text: Z/pk)

text: si

text: i

text: On

text: a

text: donc

text: Hi

text: (B¯

text: Q

text: si

text: i

text: La

text: suite

text: exacte

text: de

text: ´et

text: n

text: ´et

text: n

text: p

text: Kummer

text: donne

text: une

text: suite

text: exacte

text: longue

text: O

text: O

text: (B¯

text: (B¯

text: )pk

text: (B¯

text: Z/pk)

text: (B¯

text: G

text: n

text: n

text: ´et

text: n

text: ´et

text: n

text: m

text: (B¯

text: G

text: (B¯

text: Z/pk)

text: (B¯

text: G

text: ´et

text: n

text: m

text: ´et

text: n

text: ´et

text: n

text: m

text: Or

text: d’après

text: Lem.

text: (B¯

text: G

text: comme

text: on

text: a

text: aussi

text: Pic(B¯

text: Satz

text: ´et

text: n

text: m

text: n

text: on

text: obtient

text: (B¯

text: Z/pk)

text: ´et

text: n

text: On

text: en

text: déduit

text: que

text: (B¯

text: Q

text: ´et

text: n

text: p

text: Calculons

text: maintenant

text: Q

text: On

text: regarde

text: la

text: suite

text: exacte

text: p

text: Q

text: Q

text: l←im−

text: Q

text: n

text: p

text: p

text: n

text: p

text: n

text: n

text: Et

text: même

text: pour

text: la

text: v-topologie,

text: cf.

text: Prop

text: Si

text: X

text: est

text: un

text: espace

text: adique,

text: pour

text: nous

text: H´eit(X,Zp)

text: ←lim−H´eit(X,Z/pk)

text: et

text: H´eit(X,Qp)

text: par

text: déﬁnition.

text: Dont

text: l’existence

text: découle

text: de

text: la

text: proposition

text: mais

text: se

text: vériﬁe

text: facilement

text: dans

text: ce

text: cas

text: en

text: utilisant

text: Prop.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 ARTHUR-CÉSAR LE BRAS Les diﬀérentielles sont strictes et les Hi (X) ont donc une topologie naturelle, qui en fait dR des K-espaces de Fréchet. La topologie du dual topologique de ces K-espaces de Fréchet est la topologie localement convexe la plus ﬁne. La dernière assertion traduit le fait que la cohomologie de de Rham des aﬃnoïdes sur- convergents est de dimension ﬁnie. “ “ Lemme 3.13. — Soit K un corps p-adique (c’est-à-dire une extension ﬁnie de Q ), W un C• p K-espace de Banach et un complexe strict de K-espaces de Fréchet. Pour tout i, C• ⊗ (cid:39) C•⊗ Hi( ) W Hi( W). K K “ (cid:39) Démonstration. — On sait qu’il existe un ensemble I tel que W (cid:96) (I, K), car K est de C• “ 0 valuation discrète. Comme un complexe de Fréchets, d’après [48, §17], on a pour tout i C ⊗ (cid:39) C i W (cid:96) (I, i). K 0 On en déduit immédiatement que pour tout i C• ⊗ (cid:39) C• Hi( W) (cid:96) (I, Hi( )), K 0 d’où le lemme. On en déduit la “ Proposition 3.14. — Soit X une variété rigide lisse et Stein sur un corps p-adique K. Pour toute K-algèbre de Banach W et tout i, (cid:39) ⊗ Hi (X ) Hi (X) W. dR W dR K Démonstration. — La proposition 3.12 permet d’appliquer le lemme aux complexes des sections globales du complexe de de Rham de X et à W. Proposition 3.15. — Soit X une variété rigide lisse sur un corps p-adique K, de dimen- sion n. On a une suite exacte de faisceaux pro-étales sur X : → B → OB → OB ⊗ → · · · → OB ⊗ → 0 O Ω1 O Ωn 0. dR,X dR,X dR,X X X dR,X X X ∈ ainsi que pour tout r Z une suite exacte de faisceaux pro-étales sur X : → B → OB → − OB ⊗ → · · · → − OB ⊗ → 0 Filr Filr Filr 1 O Ω1 Filr n O Ωn 0. dR,X dR,X dR,X X X dR,X X X Démonstration. — Voir [50, Cor. 6.13]. Nous attirons l’attention du lecteur sur le fait que ‹ ‹ ‹ O ∗O ∗ la notation , Ωi désigne ici les faisceaux pro-étales ν , ν Ωi , ν étant le morphisme de sites de X X vXers X . ‹ ‹X X pro´et ´et Nous aurons également besoin du fait suivant. On dispose de morphisme de topos : −→(cid:48) −→ ν λ X ‹ X ‹ X , pro´et C,´et ´et → dont la composée est le morphisme de topos ν : X X . Le morphisme λ est induit → →pro´et ´et ∗ par le morphisme de sites X X tel que si U X est étale, λ (U) = U . Le théorème C,´et ´et C d’Elkik ([18]) donne une équivalence : (cid:39) l←im− X (cid:48) X , K ,´et C,´et (cid:48)| K K ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 20 ARTHUR-CÉSAR LE BRAS “ “ “ Remarque 3.21. — La méthode utilisée permettrait de décrire plus généralement la coho- mologie de B+ et B pour X un espace Stein lisse sur un corps p-adique. La cohomo- dR,X dR,X logie de B+ se calcule à l’aide du quasi-isomorphisme : “dR “ “ (cid:48) B O ⊗ → ⊗ − → · · · → ⊗ − Rν∗ + = B+ Ω1 t 1B+ Ωn t nB+ , dR,X X K dR X K dR X K dR évoqué dans la remarque 3.18. On déduit de cet isomorphisme l’existence d’un triangle “ “ distingué : “• ⊗ → (cid:48) B → → ⊗ − → · · · → ⊗ − Ω B+ Rν∗ + (0 Ω1 t 1B+ /t Ωn t nB+ /t). X K dR dR,X X K dR X K dR Le complexe de droite se dévisse lui-même à nouveau comme extension de → → ⊗ − − → · · · → ⊗ − − 0 0 Ω2 t 2B+ /t 1 Ωn t nB+ /t 1 X K dR X K dR • ⊗ − par Ω C( 1). L’hypercohomologie de ces complexes se calcule facilement à l’aide de la propoXsitioKn 3.14. Pour le faisceau B , sa cohomologie devrait pouvoir se calculer comme dR dans la preuve de la proposition 3.19, en choisissant un recouvrement aﬃnoïde admissible (U ) de X et une présentation surconvergente de chaque U (l’existence d’une telle pré- k k k sentation est garantie par [18, Th. 7]) : de cette façon, on peut présenter X comme limite ⊂ inverse d’espaces Stein (V ), avec pour tout k, V U . k k k Voici le résultat que l’on obtient pour X Stein lisse de dimension 1, sous l’hypothèse “ “ additionnelle que X est connexe. On a : B B H0(X , + ) = B+ ; Hi(X , + ) = 0, si i > 1 C dR dR C dR et une extension “ → ⊗ → B → ⊗ − → 0 H1 (X) B+ H1(X , + ) Ω1(X) t 1B+ /B+ 0. dR K dR C dR K dR dR (on utilise le fait que Hi (X) = 0“si i > 2, et aussi pour i = 2 puisque par“dualité de dR ∗ Poincaré“H2 (X) = H0 (X) = 0). On“a aussi : dR dR,c “ H0(X , B ) = H1“(X)⊗ B ; Hi(X , B ) = 0, si i > 1“ C dR dR K dR C dR Via ces identiﬁcations, la ﬂèche naturelle H1(X , B+ ) → H1(X , B ) se décrit comme ⊗ “ C dR C dR ⊗ → suit : sur le sous-espace H1 (X) B+ , c’est la ﬂèche évidente H1 (X) B+ ⊗ dR K ⊗dR − dR K dR H1 (X) B ; sur le quotient Ω1(X) t 1B+ /B+ , c’est la composée de la projection dR ⊗ K− dR → ⊗ − K dR dR ⊗ Ω1(X) t 1B+ /B+ H1 (X) t 1B+ /B+ avec l’inclusion H1 (X) B /B+ . Par cKonséqudeRnt, dlRe groudpRe H0(XK , B dR/B+dR) est une extensiodnR de OK(XdR)/CdRpar B /B+ . On a H1(X , B /B+ ) = CH1 d(RX)⊗dRB /t−1B+ et Hi(X , B /BC+ ) = 0 si dR dR C dR dR dR K dR dR C dR dR i > 1. L’analyse de la cohomologie de B[1/t]ϕ=1 est plus subtile. On va prouver le résultat suivant. ≥ Proposition 3.22. — Soit n 1. On note Dn le disque unité ouvert de dimension n. Pour tout i > 0, B Hi(Dn, [1/t]ϕ=1) = 0. C De même, pour tout i > 0, B Hi(An , [1/t]ϕ=1) = 0. C Dans tout ce paragraphe, I désigne un sous-intervalle compact de ]0, 1[ à extrémités des nombres rationnels. Nous allons commencer par décrire la cohomologie du faisceau B [1/t]. I Modulo une hypothèse formulée ci-dessous (preuve de la proposition 3.27) et appelée ∗ ( ) (22), nous obtiendrons des résultats plus ﬁns, sans inverser t. Ces résultats conditionnels n’interviennent pas dans la démonstration de la proposition 3.22 ; nous les mentionnons 22. Et que nous avons depuis démontrée : cf. [42, Prop. 3.11]. ##LTLine##

text: ARTHUR-CÉSAR

text: LE

text: BRAS

text: Remarque

text: La

text: méthode

text: utilisée

text: permettrait

text: de

text: décrire

text: plus

text: généralement

text: la

text: coho-

text: mologie

text: de

text: B+

text: et

text: B

text: pour

text: X

text: un

text: espace

text: Stein

text: lisse

text: sur

text: un

text: corps

text: p-adique.

text: La

text: cohomo-

text: dR,X

text: dR,X

text: logie

text: de

text: B+

text: se

text: calcule

text: à

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text: quasi-isomorphisme

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text: Prop.

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text: 28 ARTHUR-CÉSAR LE BRAS On peut désormais conclure la preuve du théorème 3.2. Démonstration du théorème 3.2. — C’est maintenant immédiat : on applique pour ﬁnir la suite exacte longue qui se déduit de la « suite exacte fondamentale faisceautique » (2) et les propositions 3.20 et 3.22 donnent que pour tout i, Hi(An , Q ) = Ker(d ) = Im(d − ). C p i i 1 Remarque 3.33. — Colmez, Dospinescu et Nizioł ont annoncé ([14]) une description de la cohomologie pro-étale de Q sur un un espace Stein lisse sur un corps p-adique, ayant p un modèle formel p-adique strictement semi-stable, en termes de la cohomologie de Hyodo- Kato de la ﬁbre spéciale de ce modèle et de la cohomologie de de Rham. Leur démonstration repose sur un théorème de comparaison entre cohomologies étale et syntomique. Dans le cas particulier de l’espace aﬃne, leur argument est présenté dans [16]. Notre démonstration est diﬀérente puisqu’elle ne fait pas appel à la théorie syntomique. Toutefois, les deux approches sont liées : en eﬀet, la relation entre cohomologies étale et syntomique peut s’interpréter en termes du foncteur décalage Lη , comme nous le montrons t dans [41, §2.8]. 4. Groupes d’extensions de certains faisceaux pro-étales L’objectif de cette section est la démonstration du résultat suivant. Théorème 4.1. — Dans la catégorie abélienne des faisceaux de groupes abéliens sur Perf , on a C,v Hom G Q Ext1 G Q Ext2 G Q a p a p a p G C 0 G C C G 0 0 a a a Q C Q Q 0 0 Q 0 0 p p p p (ces tableaux se lisant de gauche à droite). La stratégie générale pour ce faire est la suivante : T Soit G un groupe abélien dans un topos . Il existe une résolution canonique de G de longueur inﬁnie par un complexe dont chaque composante est une somme de termes de × la forme Z[Gi Zj], dont on trouvera une construction dans le style cubique dans [44] ou dans le cadre simplicial dans [36]. Soit n > 0 un entier. Si on ne s’intéresse qu’aux calculs des groupes Exti avec i < n, il suﬃt de se donner une résolution partielle de G de longueur n, i.e. un complexe C d’objets de ce topos, concentré en degrés négatifs, muni d’un → morphisme d’augmentation C G (G vu comme complexe concentré en degré 0) induisant (cid:39) (cid:54) − un isomorphisme H0(C) G et tel que Hi(C) = 0 si i = 0, n. Le triangle distingué − → → −→ H n(C)[n] C G +1 (cid:48) T montre que si G est un autre groupe abélien du topos , en appliquant à ce triangle le · (cid:48) foncteur RHom( , G ) on obtient des isomorphismes (cid:48) (cid:39) (cid:48) ExtiT (G, G ) ExtiT (C, G ) ≤ pour tout 0 i < n. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ARTHUR-CÉSAR

text: LE

text: BRAS

text: On

text: peut

text: désormais

text: conclure

text: la

text: preuve

text: du

text: théorème

text: Démonstration

text: du

text: théorème

text: C’est

text: maintenant

text: immédiat

text: on

text: applique

text: pour

text: ﬁnir

text: la

text: suite

text: exacte

text: longue

text: qui

text: se

text: déduit

text: de

text: la

text: suite

text: exacte

text: fondamentale

text: faisceautique

text: et

text: les

text: propositions

text: et

text: donnent

text: que

text: pour

text: tout

text: i,

text: Hi(An

text: Q

text: Ker(d

text: Im(d

text: C

text: p

text: i

text: i

text: Remarque

text: Colmez,

text: Dospinescu

text: et

text: Nizioł

text: ont

text: annoncé

text: une

text: description

text: de

text: la

text: cohomologie

text: pro-étale

text: de

text: Q

text: sur

text: un

text: un

text: espace

text: Stein

text: lisse

text: sur

text: un

text: corps

text: p-adique,

text: ayant

text: p

text: un

text: modèle

text: formel

text: p-adique

text: strictement

text: semi-stable,

text: en

text: termes

text: de

text: la

text: cohomologie

text: de

text: Hyodo-

text: Kato

text: de

text: la

text: ﬁbre

text: spéciale

text: de

text: ce

text: modèle

text: et

text: de

text: la

text: cohomologie

text: de

text: de

text: Rham.

text: Leur

text: démonstration

text: repose

text: sur

text: un

text: théorème

text: de

text: comparaison

text: entre

text: cohomologies

text: étale

text: et

text: syntomique.

text: Dans

text: le

text: cas

text: particulier

text: de

text: l’espace

text: aﬃne,

text: leur

text: argument

text: est

text: présenté

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text: démonstration

text: est

text: diﬀérente

text: puisqu’elle

text: ne

text: fait

text: pas

text: appel

text: à

text: la

text: théorie

text: syntomique.

text: Toutefois,

text: les

text: deux

text: approches

text: sont

text: liées

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text: eﬀet,

text: la

text: relation

text: entre

text: cohomologies

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text: et

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text: peut

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text: en

text: termes

text: du

text: foncteur

text: décalage

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text: comme

text: nous

text: le

text: montrons

text: t

text: dans

text: Groupes

text: d’extensions

text: de

text: certains

text: faisceaux

text: pro-étales

text: L’objectif

text: de

text: cette

text: section

text: est

text: la

text: démonstration

text: du

text: résultat

text: suivant.

text: Théorème

text: Dans

text: la

text: catégorie

text: abélienne

text: des

text: faisceaux

text: de

text: groupes

text: abéliens

text: sur

text: Perf

text: on

text: a

text: C,v

text: Hom

text: G

text: Q

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text: Q

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text: p

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text: C

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text: Q

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text: p

text: p

text: p

text: p

text: (ces

text: tableaux

text: se

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text: de

text: gauche

text: à

text: droite).

text: La

text: stratégie

text: générale

text: pour

text: ce

text: faire

text: est

text: la

text: suivante

text: T

text: Soit

text: G

text: un

text: groupe

text: abélien

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text: un

text: topos

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text: existe

text: une

text: résolution

text: canonique

text: de

text: G

text: de

text: longueur

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text: par

text: un

text: complexe

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text: composante

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text: Z[Gi

text: Zj],

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text: trouvera

text: une

text: construction

text: dans

text: le

text: style

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text: cadre

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text: ne

text: s’intéresse

text: qu’aux

text: calculs

text: des

text: groupes

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text: avec

text: i

text: n,

text: il

text: suﬃt

text: de

text: se

text: donner

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text: résolution

text: partielle

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text: G

text: de

text: longueur

text: n,

text: i.e.

text: un

text: complexe

text: C

text: d’objets

text: de

text: ce

text: topos,

text: concentré

text: en

text: degrés

text: négatifs,

text: muni

text: d’un

text: morphisme

text: d’augmentation

text: C

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text: en

text: degré

text: induisant

text: un

text: isomorphisme

text: G

text: et

text: tel

text: que

text: Hi(C)

text: si

text: i

text: n.

text: Le

text: triangle

text: distingué

text: H

text: n(C)[n]

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text: T

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text: que

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text: G

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text: G

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text: obtient

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text: isomorphismes

text: ExtiT

text: (G,

text: G

text: ExtiT

text: (C,

text: G

text: pour

text: tout

text: i

text: n.

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text: BANACH-COLMEZ ET FAISCEAUX COHÉRENTS SUR LA COURBE DE FARGUES-FONTAINE 31 En prenant le tiré en arrière de cette extension le long de B+ /tB+ → B /tB+ , on obtient dR dR dR dR une extension → → B → B B (cid:39) → 0 Q [1/t]ϕ=1 + /t + G 0, p dR dR a qui est celle donnée par l’application θ de Fontaine. L’identité de G dans lui-même corres- O a → pond à la fonction identité dans H0(A1 , G ) = (A1 ). La ﬂèche composée H0(A1 , G ) H0(A1 , B /tB+ ) → H1(A1 , Q ) esCt la aﬂèche quoCtient O(A1 ) → O(A1 )/C. OCn ena dé- duit quCe l’déRlémendtRId ∈ Ext1(GC , Qp ) exhibé ci-dessus corresponCd à l’extenCsion Bϕ=p de G a p a par Q donnée par l’application θ de Fontaine. p → − Pour i = 0, on a une ﬂèche Hom(C0, G ) Hom(C 1, G ), qui s’identiﬁe à a a O → O (cid:55)→ − − H0(A1 , G ) = (A1 ) H0(A2 , G ) = (A2 ) ; f(X) f(X + Y ) f(X) f(Y ). C a C C a C Son noyau est de dimension 1, engendré par f(X) = X. Ainsi Hom(G , G ) = C. a a Pour i = 2, on regarde − → − Hom(C 2, G ) Hom(C 3, G ), a a − → − Ext1(C 1, G ) Ext1(C 2, G ), a a → − Ext2(C0, G ) Ext2(C 1, G ). a a Le dernier terme est nul. Analysons le premier : la ﬂèche va de O((cid:80)A3 ) × O(A2 ) vers O ×O ×O ×O ×O C C (A4 ) (cid:80)(A3 ) (A3 ) (A2 ) (A1 ). Les applications considérées (sauf la dernière) C C C C C préservent le de(cid:18)gré. So(cid:19)it (f, g) dan(cid:18)s le n(cid:19)oyau. On éc(cid:18)rit f(x(cid:19), y, z) = k,j,l ak,j,lxkyjzl et g(x, y) = b xkyl. L’annulation sur la composante A4 donne : k,j k,l C k + j − j + l l + m a a + a = 0, k k+j,l,m j k,j+l,m l k,j,l+m (cid:18) (cid:19) (cid:18) (cid:19) (cid:54) (cid:54) pour k, m = 0 (en particulier a = 0 pour k, l > 0). Pour m = 0, on a aussi k,0,l j + l − l + m a a = 0 j 0,j+l,m l 0,j,l+m et de même en échangeant les rôles de k et m. Enﬁn a = 0. − 0,j,0 Si a = (k + 1)c (l +(cid:18)1)c (cid:19) pour k(cid:18), l > (cid:19)0, on vériﬁe immédiatement par k,1,l k+1,l k,l+1 récurrence avec la relation ci-dessus que k + j − j + l a = c c , k,j,l j k+j,l(cid:0) (cid:1) j k,j+l pour tout j > 0. On pose aussi jc = a − pour j > 0. Une récurrence immédiate 0,j 0,j 1,1 à l’aide de la relation ci-dessus donne a = j+l c pour tout l > 0. Enﬁn on pose − 0,j,l j 0,j+l c c = b . 1,kSi lesk,c1(cid:80) so1n,kt ainsi déﬁnis, alors la première ﬂèche vers A3 donne b = c − c pour j,k C j,k j,k k,j j > 1. L’autre ﬂèche donne la même égalité pour k > 1. En ﬁn de compte, si l’on pose h(x, y) = c xkyj, on a k,j k,j − − − f(x, y, z) = h(x + y, z) h(y, z) h(x, y + z) + h(x, y) ; g(x, y) = h(x, y) h(y, x). − → − En d’autres termes, (f, g) est dans l’image de la ﬂèche Hom(C 1, G ) Hom(C 2, G ). ∈ a a Enﬁn on regarde le terme du milieu. Soit ω H1(A2 , G ) = Ω1(A2 ) dans le noyau, que C a C l’on écrit ω = f(x, y)dx + g(x, y)dy. On a − f(x + y, z) f(x, y + z) + f(x, y) = 0, − − f(x + y, z) f(y, z) g(x, y + z) + g(x, y) = 0, − − g(x + y, z) g(y, z) g(x, y + z) = 0.

text: BANACH-COLMEZ

text: ET

text: FAISCEAUX

text: COHÉRENTS

text: SUR

text: LA

text: COURBE

text: DE

text: FARGUES-FONTAINE

text: En

text: prenant

text: le

text: tiré

text: en

text: arrière

text: de

text: cette

text: extension

text: le

text: long

text: de

text: B+

text: /tB+

text: B

text: /tB+

text: on

text: obtient

text: dR

text: dR

text: dR

text: dR

text: une

text: extension

text: B

text: B

text: B

text: Q

text: /t

text: G

text: p

text: dR

text: dR

text: a

text: qui

text: est

text: celle

text: donnée

text: par

text: l’application

text: θ

text: de

text: Fontaine.

text: L’identité

text: de

text: G

text: dans

text: lui-même

text: corres-

text: O

text: a

text: pond

text: à

text: la

text: fonction

text: identité

text: dans

text: G

text: La

text: ﬂèche

text: composée

text: G

text: B

text: /tB+

text: Q

text: esCt

text: la

text: aﬂèche

text: quoCtient

text: )/C.

text: OCn

text: ena

text: dé-

text: duit

text: quCe

text: l’déRlémendtRId

text: Qp

text: exhibé

text: ci-dessus

text: corresponCd

text: à

text: l’extenCsion

text: Bϕ=p

text: de

text: G

text: a

text: p

text: a

text: par

text: Q

text: donnée

text: par

text: l’application

text: θ

text: de

text: Fontaine.

text: p

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text: Hom(C

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text: Son

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text: dimension

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text: X.

text: Ainsi

text: Hom(G

text: G

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text: a

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text: dernier

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text: Analysons

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text: z)

text: k,j,l

text: ak,j,lxkyjzl

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text: L’annulation

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text: alors

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text: L’autre

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text: on

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text: k,j

text: f(x,

text: y,

text: z)

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text: z)

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text: z)

text: h(x,

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text: En

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text: f(x,

text: y)dx

text: g(x,

text: y)dy.

text: On

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text: f(x

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