TITLE: What is dispersive estimate?
QUESTION [7 upvotes]: Consider free NLS: $i\partial_tu+\Delta u=0, \quad u(0, x)=u_0$
The solution of this IVP, can be written as
$$u(x,t)=e^{it\Delta}u_0(x)$$
It is clear to me that how to prove following estimate:
$$ \|e^{it\Delta} f \|_{L^{p'}} \leq |t|^{-\frac{d}{2} \left( \frac{1}{p}-\frac{1}{p'} \right)}  \|f\|_{L^{p}} \quad (1\leq p \leq 2)$$

My question is: Why this estimate is known as dispersive estimates?

REPLY [5 votes]: To complete Piero's answer, let me give a definition of dispersiveness. To stay at a rather simple level, let me consider an linear evolution PDE with constant coefficients,
$$P(\partial_t,\nabla_x)u=0.$$
Hereabove, $P$ is a polynomial. Let me assume a property of homogeneity, $$P(\lambda^\alpha\tau,\lambda\xi)\equiv\lambda^m P\tau,\xi).$$
It is met by the wave equation and the Schrödinger operator.
When studying the wave propagation, one faces the algebraic equation
$$P(\tau,\xi)=0,\qquad\xi\in{\mathbb R}^n,\quad\tau\in{\mathbb C}.$$
This defines roots $\tau_1(\xi),\ldots,\tau_k(\xi)$ - when $\tau_j(\xi)$ is real, $\nabla_\xi\tau_j$ is a group velocity. In the best situation, they have constant multiplicities for $\xi\ne0$, and then are smooth functions. Then the equation is dispersive if none of these functions is linear, meaning that the velocities $\nabla\tau_j$ do vary with $\xi$.
Because $\tau(\lambda\xi)=\lambda^\alpha\tau(\xi)$ for $\lambda>0$, every equation whose order in time differs from the order in space ($\alpha\ne1$) is dispersive - mind however that if $\alpha<1$, then the Cauchy problem is not even well-posed. But even if $\alpha=1$, the equation can be dispersive, as shown by $\Box u=0$. Other examples come from linearized gas dynamics: the acoustic waves satisfy $\tau(\xi)=v\cdot\xi\pm c|\xi|$ ($v$ the fluid velocity, $c$ the sound speed) and thus are dispersive, though the entropy waves satisfy $\tau(\xi)=v\cdot\xi$, hence are not dispersive.
The fact that dispersive equations satisfy $L^p-L^q$ decay estimates is an extension of so-called Strichartz estimates.

REPLY [4 votes]: The intutitive picture is the following: there is a certain amount of "mass", which at time $t=0$ you might imagine concentrated in a heap near a location $x=0$. This heap evolves with time according to a differential equation.
If the equation is a transport equation or a 1D wave equation (or the solution is a soliton for some suitable nonlinear equation), the heap moves in some direction but remains concentrated near some point $x(t)$. This behaviour is non dispersive.
But if the equation is of dispersive type, the heap also spreads as time evolves: the total mass will remain the same, by conservation of mass, but it will spread on a larger and larger region of space. This is a dispersive behaviour.
In your example, if you take $p=2$ you have conservation of mass, measured in $L^2$ norm. However, if you take $p=\infty$, you see that the "peak" of the heap becomes lower, at a rate of $t^{-n/2}$.
If you take as initial data a wave packet, you can actually see this effect: the packet moves at a speed proportional to the frequency (as postulated by quantum mechanics), it spreads (this is called decoherence in physics), and its height decreases precisely at the stated rate.