Conservation laws and compact support

Question

Assume we have $F \in C^2(\mathbb{R})$ and $u_0 \in L^\infty(\mathbb{R})$ with sompact support. I am wondering whether the entropy solution to the Cauchy Problem on $(0,T) \times \mathbb{R}$: $$ \partial_tu(t,x)+\partial_x F(u(t,x))=0, \quad u(0,\cdot)=u_0 $$ is compactly supported as well. Intuitively, one might expect that the initial datum ist transported, i.e. support indeed remains compact. Does anyone have a reference/proof/counterexample?

Thank you!

Answer 1

Likely yes. I wonder, does the following help you to find a reference?

[Andreianov...] say:

"The best-known case is A=0,... " (I think this is your case) "... Via the method of doubling variables, these entropy inequalities imply Kato’s inequality: for any couple (u, v) of entropy solutions $$ ∂_t|u − v| + div_x ( sgn(u − v)(f (u) − f (v)) ) ≤ 0 \tag2 $$in the sense of distributions (i.e. with compactly supported smooth test functions), where sgn(.) denotes the sign function. The finite speed of propagation (for bounded f ′) is used in [32] to construct appropriate sequences of test functions to be inserted into Kato’s inequality (2) in order to infer uniqueness. As a matter of fact, the speed of propagation is the key heuristic issue in the discussion below...."

That talking about finite speed of propagation is promising. (Thought the term might be misleading.)

They talk more about the pure hyperbolic local case (A = 0) later and I think they circle around your topic :

"... The possible non-uniqueness demonstrated by counterexamples due to Panov may be explained by the infinite speed of propagation caused by the unboundedness of f ..."

I hope that the works they cite can you lead to something like that if $F$ has (a certain degree of Holder) continuity, you have a positive result.

Andreianov, Boris; Brassart, Matthieu, Uniqueness of entropy solutions to fractional conservation laws with “Fully infinite” speed of propagation, J. Differ. Equations 268, No. 7, 3903-3935 (2020). ZBL1473.35619. (HAL)