Question

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

Answer 1

Hille-Yosida existence theorem theorem is one example of abstract evolution equation. In its simplest form it starts with a closed, densely defined operator $A$ on a Hilbert space $H$ with domain $D(A)$ such that

$$(Au, u)\geq 0,\;\;\forall u\in D(A) $$

and $\boldsymbol{1}+ A: D(A)\to H$ is surjective. (Such operators are called linear maximal monotone operators.) The operator $A$ is typically unbounded. We can make $D(A)$ into a Banach space with norm

$$\Vert u\Vert_A:= |u|+ |Au|,\;\;\forall u\in D(A), $$

where $|-|$ denotes the norm in $H$.

The Hille-Yosida theorem states that for any $u_0\in D(A)$ there exists a unique function

$$ u\in C^1\bigl(\;[0,\infty), \; H\;\bigr)\cap C^0\bigl(\; [0,\infty), \;D(A) \;\bigr)$$

satisfying the initial value problem

$$\frac{du}{dt}+ Au=0,\;\; u(0)=u_0. $$

If additionally $A$ is symmetric, i.e., $(Au,v)=(u,Av)$, $\forall u,v\in D(A)$, then one can relax the assumptions on $u_0$. For more details see H. Brezis' book Functional Analysis, Sobolev spaces and Partial Differential Equations.

How hard is it to apply this in practice? The surjectivity of $ \boldsymbol{1}+ A$ is the great obstactle because it is in essence an existence result: for any $v\in H$ one can find $u\in D(A)$ such that

$$u+ Au= v. $$

This is nontrivial and requires typically elliptic theory. The domain $D(A)$ of $A$ is often difficult to identify. If for example $A$ is an elliptic operator of the form

$$A= -\sum_{1\leq j} \partial_{x_j}\bigl( a_{ij}(x) \partial_{x_i} \bigr), $$

where $a_{ij}(x)$ are smooth functions on a bounded domain with smooth boundary, then the domain of $A$ depends on the boundary condition you impose. For a Dirichlet condition the domain will be

$$ D(A)= \bigl\lbrace u\in H^2(\Omega);\;\;u|_{\partial \Omega}=0\bigr\rbrace. $$

For more general elliptic operators, the domains could be more complicated. Let me point out that the fact that $A$ defined by the above differential operator with the above domain is a closed operator is already a nontrivial fact. It ultimately relies on a priori elliptic estimates for elliptic boundary value problems. In their complete generality they represent a major mathematical achievement of the 20th century.

One can work on Banach spaces as well, and more importantly, the Hille-Yoshida theorem has a nonlinear and highly nontrivial generalization. On this topic I recommend V. Barbu's book Nonlinear Differential Equations of Monotone Types in Banach Spaces for general abstract nonlinear results and many applications.

Answer 2

I have two examples:

One example can be found in the wave equation, $u_{tt} - u_{xx} = 0$. Let the domain be $\mathbb{R}$ and the initial data be $\sin (x)$. Then for all time the solution will not be in $H^1$. Rather, the solution is in $C^{\infty} \cap L^{\infty}$.

Another even more complicated example is when you consider gradient flows on $\mathcal{P}_2$, the space of borel probability measures of bounded second moment. The example I have in mind are gradient flows of the Interaction Energy, where a distribution can go in and out of the Sobolev spaces you're referring to, even to dirac delta point masses.

Edit: On the second example, I think this paper most directly answers your question, in particular from the abstract:

"... The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. ..."

That's just the first example to come to mind, I'm sure there are others.

Answer 3

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.