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I'm now working on somehow an inverse problem of an ODE:

Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t).

Now there is a bounded, simply connected open subset $\Omega\subset \mathbb{R}^{n}$, define for $a,b\in\Omega$ the space:

$$M(a,b) = \left\{f\in C^{\infty,1}_{L}(\Omega): x_{f,a}(T) = b\text{ and } x_{f,a}(t)\in\Omega,\forall t\in[0,T]\right\}$$

Where $C^{\infty,1}_{L}$ stands for Lipschitz continuous and smooth vector-valued functions satisfying

$$\sup \left\vert \frac{\partial^{k}_{x_{i}}f(x)-\partial^{k}_{y_{i}}f(y)}{x-y}\right\vert\leq L,\forall k\geq0$$

Apparently $M(a,b)$ is not a Hilbert space (not even a linear space), so many functional analysis tools fail for analysis on $M(a,b)$.

My question is, for any $f_1,f_2 \in M(a,b)$, is there any bound of $\Vert f_1-f_2\Vert_{W^{s,p}(\Omega)}$? Or is there any related research?

----Update 2024/09/09

I've found a article which may help: Vrabie, Ioan I. “Compactness of the Solution Operator for a Linear Evolution Equation with Distributed Measures.”, probably i've made the question complicated.

Thx.

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  • $\begingroup$ What do you mean by max Hölder coefficient $L$? A map $f$ is Hölder if it satisfies $|f(x)-f(y)|\le C|x-y|^\alpha$ for given $C$ and $\alpha$. You mean to fix both, don't you? $\endgroup$
    – Pietro Majer
    Commented Sep 8 at 9:39
  • $\begingroup$ I mean $\sup \vert f(x)-f(y)\vert / \vert x-y\vert \leq L$, it should be Lipschitz though, i'll edit my question, thx for point me out. @PietroMajer $\endgroup$
    – Sqr
    Commented Sep 8 at 12:30
  • $\begingroup$ What $f_1$ and $f_2$ are? $\endgroup$
    – Alexandre Eremenko
    Commented Sep 8 at 13:01
  • $\begingroup$ For any $f_1$ and $f_2$ s.t. $f_1,f_2\in M(a,b)$, i've edited the question, thx.@AlexandreEremenko $\endgroup$
    – Sqr
    Commented Sep 8 at 13:12

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