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



abstract evolution equations
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.


