# Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} It is well known that if $f$ is continuous and furthermore Lipschitz continuous in an appropriate sense, then there exists a unique solution $y \in C^1([0,T], \mathbb{R}^n)$ satisfying the differential equation at every point of the time interval $(0,T)$ (Picard–Lindelöf).

The question: Which general assumptions on $f$ ensure solvability in a (Sobolev) weak sense? In particular, I want the solution $y$ to be in $H^1 = W^{1,2}$, i.e. $y$ should be $L^2$, $y$ should be weakly differentiable, $y'$ should be $L^2$ and the weak derivative of $y$ should coincide with $f(t,y(t))$ almost everywhere.

This should be an extremely basic question, but I do not know relevant literature.

[I seem to have an existence proof in the case $f(t,y) = A(t) y$, where $A \in L^2([0,T], \mathbb{R}^{n \times n})$, by mimicking the usual proof of the Picard–Lindelöf theorem and simply considering the usual integral operator as an operator $L^2([0,T], \mathbb{R}^n) \to L^2([0,T], \mathbb{R}^n)$. This case is particularly important to me. Note that here, $f(t,y)$ is not continuous in $t$.]

• You may see this: math.stackexchange.com/a/460809/254733 Commented Mar 10, 2016 at 14:27
• Also, a weak formulation for this first order equation would involve $L^2$ as a trial and test space, so maybe it is more or less the same as your second method with the integral operator from $L^2\to L^2$ Commented Mar 10, 2016 at 14:29
• @Svetoslav: Thanks! I'll check that. (Sorry I didn't reply sooner -- didn't get a notification of your comment, even though of the bounty.) Meanwhile, I stumbled on Carathéodory's existence theorem (which was curiously absent from by education) and which might also provide an answer. It is formulated for absolutely continuous functions instead of Sobolev functions. Commented Mar 15, 2016 at 23:29
• @IngoBlechschmidt By the way, you may know, but just to mention that in $1D$ all Sobolev functions are absolutely continuous. In dimension $\ge 2$ they are absolutely continuous on almost all lines parallel to the coordinate axes (the ACL characterization). Commented Mar 16, 2016 at 7:22

This is mostly for anyone who stumbles upon this question and wants a more detailed answer. There are also partly answers in the question itself and in the comments. I try to extend it a bit with references and give a typical application for this formalism which arises in the study of evolutionary PDEs.

We want to investigate the ODE $$\begin{cases} \dot y(t)=f(t,y) \\ y(0)=y_0. \tag{\star} \end{cases}$$ Your aim is to equip $f$ with conditions such that the ODE has a solution $y \in H^1(0,T;\mathbb{R}^d)$ almost everywhere.

The key-word was already given: Carathéodory's theorem

It is the extension of Peano's existence theorem to ODE with discontinuous right hand side. It gives the existence of a solution in a Sobolev space. A short version can be found in "Nonlinear Partial Differential Equations with Applications" by Tomas Roubicek.

Theorem (Carathéodory). Let $T$ be fixed and $f: (0,T) \times \mathbb{R}^k \to \mathbb{R}^k$ such that $f(t,\cdot) \in C(\mathbb{R}^k)$, $f(\cdot,r) \in L^1(0,T)$ and $|f(t,r)| \leq \gamma(t)+C|r|$ for some $\gamma \in L^1(I)$. Then ($\star$) has almost everywhere a solution $y \in W^{1,1}(0,T;\mathbb{R}^k)$. Moreover, if $f(t,\cdot)$ is Lipschitz continuous, then the solution is unique.

• Here $f(\cdot,r)$ indicates the mapping $t \mapsto f(t,r)$ for fixed $r$ and analogously for $f(t,\cdot)$
• Other sources are:

• "Finite Element Methods for Incompressible Flow Problems" by Volker John
• "Ordinary Differential Equations" by Wolfgang Walter
• "Ordinary Differential Equations" by Jack Hale
• We come back to a comment who mentions the ACL characterization. Since $y \in W^{1,1}(0,T;\mathbb{R}^k) \hookrightarrow C([0,T];\mathbb{R}^k)$ and $y' \in L^1(0,T,\mathbb{R}^k)$ we have $y \in AC([0,T];\mathbb{R}^k)$

• Now you want a solution in $H^1$. This can be easily achieved if we additionally assume that $f(\cdot,r) \in L^2(0,T)$.

• Probably the main application for this theorem is in the existence of weak solutions of evolutionary PDEs when doing a Galerkin approximation. There we reduce the infinite-dimensional problem to a finite-dimensional one and reduce the PDE to an ODE. Mostly the right hand side is in a space such as $L^2(0,T;H^{-1})$ and classical theorems like Peano are not applicable.

• Example: We consider the parabolic PDE $y'+Ay=f$ on $\Omega \times (0,T)$ with homogeneous Dirichlet boundary and data $f\in L^2(0,T;H^{-1})$, $y_0 \in L^2$. $A$ is typically a second order partial differential operator in divergence form. The weak form reads $\langle y',v \rangle_{H^{-1}} + a(y,v)= \langle f,v \rangle_{H^{-1}}$ for all $v\in H_0^1$ where the bilinear form $a$ corresponds to $A$. The finite dimensional problems reads $\langle y_k',v \rangle_{H^{-1}} + a(y_k,v)= \langle f,v \rangle_{H^{-1}}$ for all $v \in V_k$ where $V_k$ is just that $\cup_k V_k$ is dense in $H_0^1$. With the Galerkin ansatz this problem reduces to an ODE with the vector-valued right hand side $F(t)=(\langle f(t),v_j \rangle_{H^{-1}})_j \in L^2(0,T;\mathbb{R}^k)$. Thus Carathéodory's theorem gives a solution $y_k \in H^1(0,T;V_k)$ to the Galerkin problem. In the end one derives energy estimates and get that a subsequence of $y_k$ converges to a $y$ which is a weak solution to the original PDE. For a reference see for example "Optimization with PDE Constraints" by Hinze, Pinnau, Ulbrich.