Large-time behavior of solutions to a scalar conservation law in several space dimensions

Abstract:

We consider solutions of the Cauchy problem in ${\mathbf {R}}_ + ^{n + 1}$ for the equation ${u_t} + {\operatorname {div}_x}f(u) = 0$. The initial data is assumed to be a compact perturbation of a function of the form, $\varphi (x) = a$ for $\left \langle {x, \mu } \right \rangle > 0$, $\varphi (x) = b$ for $\left \langle {x, \mu } \right \rangle < 0$, where $a$ and $b$ are constants and $\mu$ is a given unit vector. The Cauchy problem together with an entropy condition on $u$ is known to be well posed. The solution with unperturbed initial data, $\varphi (x)$, is a traveling shock, $\varphi (x - \overrightarrow k t)$, provided that $\varphi (x - \overrightarrow k t)$ satisfies the entropy condition (an inequality on $a, b, \mu$, and $f$). Assuming this type of condition on $\varphi$, we study the large-time behavior of $u$. In particular, we show that $u$ converges to a traveling shock whose profile agrees with $\left \langle {x, \mu } \right \rangle = 0$ outside of a compact set.