Existence theorem of weak solutions of $u_t+f(u)u_x=0$

Question

Consider this PDE:

$\begin{cases}u_t+f(u)u_x=0\\ u(x,0)=\varphi(x)\end{cases}$

Has this PDE weak solutions whatever is $f$ or $\varphi$? I want to find an existence theorem and bibliography about that?

Can anyone help me?

Thanks in advance!

@KurtG. , yes. I 'm trying to solve something like this (https://math.stackexchange.com/questions/4598413/burgers-pde-with-piecewise-constant-initial-condition). I am in between of not existence and something like physical solution. — Kώστας Κούδας, Dec 22 '22 at 19:26

EditPiAf · Answer 1 · 2022-12-23T13:41:05.873

Introduce $F(u) = \int_0^u f(\theta)\, \text d\theta$. This way, the transport velocity is $f(u) = F'(u)$, and the PDE reads $u_t + F(u)_x=0$ in conservation form. Using the definition of weak solutions (see e.g. Wikipedia and the books by R.J. LeVeque), one might be able to prove that the strong solution defined implicitly by $$ u = \varphi(x - f(u)\, t) $$ is also a weak solution of the initial-value problem, as long as the solution remains smooth. Unfortunately, for discontinuous solutions, weak solutions might not be unique. It is then necessary to add further requirements on weak solutions to select the physically correct one, among those called entropy solution or vanishing-viscosity weak solutions.

Thanks for your answer! Happy holidays! I know that weak solutions might not be unique. I didn't know if they always existed. I believed that sometimes strong solutions do not exist and neither do weak solutions. — Kώστας Κούδας, Dec 23 '22 at 17:00

score 1 · Answer 2 · answered Dec 23 '22 at 13:33

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There is a section on weak solutions to the above problem on Evans' book. Check "Partial Differential Equations", Evans, Lawrence C., section 3.4. There is even more stuff in chapter 11. You need some hypotheses on $f$ and $\varphi$ though to use that theory.

answered Dec 23 '22 at 13:33

Lorenzo Pompili

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Thanks for your answer and Merry Christmas! I had downloaded Evans' book, but I haven't read it yet. I was reading Strauss' book and Pinchover&Rubinstein. I will read your recommendation more carefully. – Kώστας Κούδας Dec 23 '22 at 16:56

score 1 · Accepted Answer · answered Dec 23 '22 at 19:04

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Finally, I found the answer here:

Thanks for the time you spent on my question!

answered Dec 23 '22 at 19:04

Kώστας Κούδας

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Existence theorem of weak solutions of $u_t+f(u)u_x=0$

3 Answers3