Consider the initial value problem for scalar conservation laws
$$\begin{eqnarray} u_t+f(u)_x=0\\
u(x,0)=u_0(x) \end{eqnarray}$$
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If $u_0 \in L^{\infty}(\mathbb{R})$ we have $\Vert u(.,t)\Vert_{\infty} \leq \Vert u_0\Vert_{\infty}$
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If $u_0 \in L^{1}(\mathbb{R})$ we have $\Vert u(.,t)\Vert_{1} \leq \Vert u_0\Vert_{1}$ (which follows from the contraction principle)
Do we have such results for other values of $p \in (0,\infty)$ ? If so how to prove it?
Any comments or references would be greatly appreciated.
Best Answer
Let $u\in C([0,T];L^{1}(\mathbb{R}))$ be an entropy solution. Then for each convex entropy function $\eta,$ the entropy condition implies the following. $$\int\limits_{\mathbb{R}}\eta(u(x,t))dx \leq \int\limits_{\mathbb{R}}\eta(u(x,0))dx.$$ Choosing $\eta(u)=u^p$ we get the desired $L^p$ bounds on the entropy solution.
P.S.: However, for weak solutions in general such $L^p$ bounds do not exist for any $p \in (0,\infty].$