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Denote $P(\tau,\eta,\xi):=\det L(\tau,\eta,\xi)\in{\mathbb R}[\tau,\eta,\xi]$. This is a homogeneous polynomial of degree $N$, whose leading term in $\tau$ is just $\tau^N$. Let us fix $j$, so that $\ …

This is known as the Goursat problem, because the boundary condition are given over two intersecting characteristic lines. Notice the necessary condition (which turns out to be sufficient): $$c_1(0)=c …

I am Denis Serre, one of the authors, and am happy to give you an answer. My impression is that everything is OK. The matter is to see the equivalence between the expressions $$A:=\sum_{|\alpha|\le s} …

My impression is the following. In order that $u$ be an entropy solution, it must not have increasing shocks (because $f$ is strictly convex). Thus if $u$ does not experience at $(t,x)$ a decreasing d …

This is not a transport equation. It is a conservation law. The difference between these class is that a TE is of the form $\partial_tu+a(t,x)\cdot\nabla_xu=0$, for which the essential supremum/infimu …

I think that you should read L. Gaarding's seminal paper Linear hyperbolic partial differential equations with constant coefficients, Acta Math 85:1-62 (1951). It explains why hyperbolicity is the app …

You made a confusion between the symbol (here $-(1+\xi^2)\tau^2+\xi^4$), and the principal symbol, which gathers the monomials of highest degree. Since the latter is $\xi^4-\xi^2\tau^2$, which splits …

They are not. First of all, the existence of a convex entropy is not meaningful for a system given in this quasi-linear form. The reason is that you might make a change $v=\phi(u)$ of unknown, but the …

The answer to this question depends a lot on the space dimension $n$. It is true that if $n=1$, the Cauchy problem has been studied with data in either $L^\infty(R)$ or $BV(R)$. For superlinear wave e …

The method of characteristics for $2\times2$ systems is discussed by C. Dafermos in his book. In the third edition, it is Chapter XII. However, I want to make a few points: as mentionned by Willie, …

Unless an evolution PDEs be linear, a uniqueness proof is always nonlinear in essence: you prove that some distance $d(u(t),v(t))$ between two solutions is bounded in terms of $d(u(0),v(0))$. To carr …

The regularity mentionned in the theorem is not accurate, for two reasons. The first is that the spaces ${\cal C}^k$ don't behave well with PDEs. Often, it is better to work with ${\cal C}^{k,\alpha}$ …