论文标题
标量保护法的长期行为与严重耗散
Long-time behavior of scalar conservation laws with critical dissipation
论文作者
论文摘要
关键的汉堡方程$ \ partial_t u + u \ partial_x u +λu= 0 $是用于运输与扩散之间在流体中发生冲击的竞争的玩具模型。众所周知，光滑的初始数据不会在有限的时间内产生冲击。关于“类似冲击”的初始数据的长期行为知之甚少：$ u_0 \ to \ pm a $ as $ x \ to \ mp \ infty $。我们在多维临界标量保护定律的一般设置中描述了这种长期行为，$ \ partial_t u + \ text {div} f（u） +λu= 0 $当初始数据在无穷大处有限制。渐近溶液给出了渐近性，我们以最佳扩散速率证明了它们的稳定性。
The critical Burgers equation $\partial_t u + u \partial_x u + Λu = 0$ is a toy model for the competition between transport and diffusion with regard to shock formation in fluids. It is well known that smooth initial data does not generate shocks in finite time. Less is known about the long-time behavior for `shock-like' initial data: $u_0 \to \pm a$ as $x \to \mp \infty$. We describe this long-time behavior in the general setting of multidimensional critical scalar conservation laws $\partial_t u + \text{div}f(u) + Λu = 0$ when the initial data has limits at infinity. The asymptotics are given by certain self-similar solutions, whose stability we demonstrate with the optimal diffusive rates.
