Minimal assumptions for existence of solutions of First order PDE

Question

I'm looking for a reference about existence of linear homogeneous first order PDE, in particular about the minimal assumption on the data. In literature I found that one require $C^1$-regularity on the data to prove existence and uniqueness. I wonder if I can relax this condition ($C^{0}$ or $C^{0,\alpha}$) to obtain again the existence of solutions (the uniqueness will be lost). Maybe this problem is quite standard, but I haven't found anything about it.

Consider the coordinates $(x,y)\in\mathbb{R}^{n}\times\mathbb{R}^{m}$. Given a vectorial function $F:\mathbb{R}^{n+m}\to \mathbb{R}^{n+m}$ and a function $g(x) \in C^1 $, I'm looking for a function $u$ which solves (locally) $$ \begin{cases} \nabla u\cdot F=0\\ u(x,0)=g(x), \end{cases} $$ considering that the following transversality condition is satisfied $$ F\cdot e_{y_j}\not=0, \quad \text{for every } j=1\dots, m. $$

The first thing that I want to note is that the boundary condition is on a set lower dimension. I think that this is better for finding solutions: we can impose a more stronger boundary condition, e.g., $u(x,y_1,\dots,y_{n-1},0)=g(x)$.

My question is about the minimal regularity assumptions on $F$ to find a solution.

I know that if $F\in C^1$, then we have (local) existence and uniqueness for our problem. This theorem can be found in the Evans's book for example.

If we only suppose that $F$ is continuous, the existence of solutions for the problem is guaranteed? I'm pretty sure that one lose uniqueness of solutions.

My guess is that one can find solutions. The strategy used to solve this problem is to find the characteristic curves by solving an ODE system and to applying the inverse function theorem.

Let us consider $(s,\sigma)\in \mathbb{R}^{n+m-1}\times\mathbb{R}$ and solves $$ \partial_\sigma x(\sigma,s)=F(x),\\ x(s,0)=(s,0), $$ which admits solution if $F$ is continuous by Peano's existences theorem. One has that the function $x=x(s,\sigma)$ satisfies $|J x (0)|\not=0$ and $x(0)=0$, by the transversality assumption and $J x$ has the same regularity of $F$, that is $x \in C^1$. So, we can apply the inverse function theorem to find that there exists the inverse map $x^{-1}$ of class $C^1$. Then, by construction, the function $g\circ x^{-1}$ is a $C^1$ solution to our problem.

Any suggestions or references are welcome!

Answer 1

There are two a priori different points of view. The first one is the Lagrangean where you look at the ODE $$ \dot x(t)=X(x(t)), $$ where $X$ is your given vector field. Then, as you mention, a local Lipschitz continuity assumption of $X$ is enough to ensure existence, uniqueness and well-posedness. Similar statements are true in an infinite dimensional Banach framework. In finite dimension, continuity will ensure existence via the Peano theorem, although that result cannot be extended to an infinite-dimensional situation: you will find some counterexamples of this in Dieudonné's treatise Elements of Analysis. You can slightly relax that Lipschitz assumption to an Osgood modulus of continuity such as $r \ln r$ (see for instance the Bahouri-Chemin-Danchin book).

The second point of view is the Eulerian perspective where you look directly at the PDE associated to your vector field. Following the works of DiPerna & Lions, it is not difficult to prove existence for solutions of the PDE associated to $X$ when $X$ has $L^\infty$ coefficients, but their main point is that uniqueness of weak (bounded) solutions occurs as well if you assume that your vector field is $W^{1,1}$ with null divergence. This was generalized by Ambrosio who proved the same uniqueness result for $X$ in $BV$, still assuming null divergence. The null divergence assumption was relaxed by Bianchini & Bonicatto later on when they proved a conjecture by Bressan for the so-called nearly incompressible vector fields, which means essentially that there exists a positive $L^\infty$ function $\alpha$, such that $1/\alpha$ is also bounded, and $$ \text{div} (\alpha X)=0. $$ Last but not least, Ambrosio showed that the Eulerian and Lagrangean points of view are equivalent, via simple generic assumptions on $X$.

I add below a response to a comment, too long to fit the comments requirements. @Simmetrico It is a classical fact that the flow of the autonomous $\dot \psi=F(\psi)$ is of class $C^k$ whenever $F$ is of class $C^k$ for $k$ integer $\ge 1$ and a similar statement is true for the Lipschitz case ($F\in C^{0,1}$ implies that the flow is $C^{0,1}$, at least with respect to the initial condition). For the $C^{0,\alpha}$ case (Hölder), no uniqueness could be expected; for the $C^{k,\alpha}$ case with $k\ge 1$, I am afraid that you have to follow the classical proof to get the fact that the flow is also $C^{k,\alpha}$. The classical result boils down to a result of continuous dependence of the ODE with respect to some parameters. I guess in fact that it is possible to review the proof for $F$ in a Besov space $B^s_{\infty, \infty}$ where $s\ge 1$ or a Triebel-Lizorkin space with the same homogeneity. As mentioned above in the first part of my answer, there is a logarithmic space beyond the Lipschitz assumption. It should be noted that the Osgood modulus property was not completely incorporated into the generalizations of the DiPerna-Lions theory. The Bahouri-Chemin-Danchin book is tackling some examples related to some vector fields solutions of the Euler equation which are not included in the DiPerna-Lions theory.

Answer 2

I give an answer.

The condition $F\in C^1$ guarantees the uniqueness of solutions to the ODE system, namely, the characteristic curves. Using this, one can show that the map $x(s,\sigma)$ is $C^1$ in the $s$-variable, by differentiable dependence of the initial data (I don't remember where I see that, but I'm pretty sure of this) and $C^2$ in the $\sigma$-variable by construction. So, the inverse function theorem applies and one can conclude that $u\circ x^{-1}$ is a $C^1$ (local) solution to the problem.

If we suppose that $F \in C^0$ or $F\in C^{0,\alpha}$ we don't have uniqueness for the ODE system and we cannot have differentiable dependence of the initial data. Hence, in general, the map $x \not\in C^1$ and we cannot conclude.