3
$\begingroup$

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some additional conditions ($(I+\epsilon A)$ is one-to-one onto a subspace of the range and the inverse of $(I+\epsilon A)$ is a contraction).

This theory uses discretization of the term $\frac{du}{dt}$ and the Crandall-Liggett theorem. The above conditions on $A$ give convergence of the discretization.

My questions:

  1. Is there an existence proof for such a theorem where time discretization is not used?

  2. Is there any some similar such existence result for equations $$F(u) + Au = f$$ where $F$ is a some operator that is not $\frac{d}{dt}$? I know this is a vague question but if I can't "put $F$ into $A$" then is there a way to solve this?

  3. Suppose that $A=A(t)$ depends on time. I have found literature (which is quite old now) that really requires the domain of $A(t)$ to be independent of $t$: $D(A(t)) = D(A(0))$ for the results to carry through. I ask if I am missing some recent work in which the conclusions of the Crandall-Liggett are valid with minor adjustments to the premises of the theorem for time-dependent operators $A(t)$ with domains not time-independent?

My first two questions stem from the fact that I don't have $\frac{d}{dt}$ but a different operator there. I don't understand fully the necessity of $\frac{d}{dt}$...

Improve this question
$\endgroup$
4
  • $\begingroup$ Dear @TheBook: How does the tag 'ap.analysis-of-pdes' apply to this question? $\endgroup$
    – Ricardo Andrade
    Commented Apr 1, 2014 at 17:14
  • $\begingroup$ @RicardoAndrade Well this certainly has applications to PDE. I need this result for a PDE I am considering and thought other people working in PDE might be knowledgeable on this topic. $\endgroup$
    – TheBook
    Commented Apr 1, 2014 at 17:47
  • 1
    $\begingroup$ @TheBook: Is it a secret or can you tell us more about your operator $F$? We cannot really speculate otherwise... $\endgroup$
    – András Bátkai
    Commented Apr 1, 2014 at 20:52
  • $\begingroup$ @AndrásBátkai In my case the equation $F(t,u(t)) + A(t)u(t) = f(t)$ holds in a Banach space $X(t)$ (parametrised by time) and $F$ is a sort of a generalised time derivative. The details are a bit lengthy so that's why I left it out. $\endgroup$
    – TheBook
    Commented Apr 2, 2014 at 14:51

2 Answers 2

Reset to default
3
$\begingroup$

I do not think it is good to ask three questions in one, since I will only address the first now.

I would put the question back and ask you: how do you define a "solution"? Usually you have an existence proof of some object and if they are good enough, call them generalized solutions. If everything is perfect, you can try to prove some nicer properties to get back "classical" solutions, but in many instances you really need generalized solutions (viscosity solutions, shocks, etc.).

In the Crandall-Liggett theory generalized solutions are defined as functions which can be approximated by discretizations. Hence I do not know of and would be surprised if there would be any proof which do not use discretizations at some point.

This may work, however, for something different than the time derivative, if it retains some structural properties.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Thanks for the answer. I see now that a mild solution is defined as the limit of the discretised solutions as you said. My intention is to obtain such a mild solution and then use some properties of my particular problem to show that it is a weak/strong solution too. If I have the equation $F(u) + A(u) = f$ as in the OP, let me write instead $u_n - u_{n-1} + A(u_n) = f$. Now this is an elliptic problem, and if I have appropriate properties on $A$ and $f$, I can use Crandall-Liggett. I will obtain a limit function $u$. The question is whether $u$ solves the PDE in some sense... $\endgroup$
    – TheBook
    Commented Apr 2, 2014 at 14:49
  • $\begingroup$ ..My understanding is, if the right hand side $F(u) = -A(u) + f$ is nice enough, then it ought to. Does that make sense? Maybe this is what you meant by structural properties. I mean I don't see anything special about the time derivative... $\endgroup$
    – TheBook
    Commented Apr 2, 2014 at 14:49
2
$\begingroup$

Commenting on what András has already written: While Crandall-Liggett's theory relies upon time discretization, Galerkin-type results often rely upon space discretization and, sometimes, they have sets of assumptions comparable with Crandall-Liggett's accretivity. The notions of solution are different, though, although they agree in some relevant cases (like linear equations :) ).

The book of Lions from 1961 is a standard reference for a functional analytic approach to Galerkin's method.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .