TITLE: abstract evolution equations
QUESTION [8 upvotes]: Hi
Whenever I read a book on evolution equations, they set up, say the parabolic PDE
$$\dot{y} = Ay + f$$
in abstract function spaces (eg. $L^2(0,T;V)$ and $L^2(0,T;V^*)$). In examples, they always pick $V = H^1$. Can anyone give me an example where $V$ is chosen to be something other than $H^1$. Does it always need to be something like $H^k$? What's the point of calling this "abstract" if we only look at Lebesgue/Sobolev spaces (and/or $C^k$ spaces)? What's an example of a truly abstract equation?
Thanks

REPLY [4 votes]: In my opinion, the adjective "abstract" refers to the fact that one define mild solution by applying a density argument, and these mild solution are not classical solutions. Let us go back to the context of Hille-Yosida theorem, explained in Liviu's answer. If $u_0\in D(A)$, there is nothing abstract, the solution is continuous differentiable with respect to time, with values in the pivotal Hilbert space $H$ (often $H=L^2$). But if the operator is maximal monotone, the fact that $S_t:u_0\mapsto u(t)$ is linear and contracts the $H$-norm allows us to extend $S_t$ to data $u_0\in H$; in this case, the solution is not time-differentiable and teh solution is not more than continuous as an $H$-valued function. One speaks of a continuous semi-group. An even more important fact is that the non-homogeneous equation
$$\frac{du}{dt}=Au+f$$
can be solved, again in an abstract sense, by applying the Duhamel principle
$$u(t)=S_tu_0+\int_0^tS_{t-s}f(s)ds.$$
Of course, this linear theory has a non-linear counterpart. But then something new happens. While functional analysis often yields simultaneously existence and uniqueness in the linear case (well-posedness, including the continuous dependence upon data), the mild solutions of non-linear evolution equations are not always unique. A striking example is that of the Navier-Stokes equation governing the motion of an incompressible fluid in the three-dimensional domain. The gap between existence and uniqueness is worth a Million Dollars.