论文标题
与时间相关阻尼的可压缩欧拉方程的奇异性形成
Singularity formation for compressible Euler equations with time-dependent damping
论文作者
论文摘要
在本文中，我们将带有时间依赖性阻尼\ frac {\ a} {（1+t）^λ} u的压缩欧拉方程在一个空间维度中。通过为平滑解决方案构建“解耦” Riccati类型方程，我们提供了一些足够的条件，在这些条件下，经典解决方案必须在有限的时间内分解。作为副产品，我们表明衍生物爆炸，有点像冲击波的形成，如果初始数据的衍生物即使在阻尼系数到具有代数增长速率的无限时，初始数据的衍生物也适当大。此外，我们分别研究了λ\ neq1和λ= 1，此外，我们的结果对溶液的大小和初始Riemann不变性的阳性/单调性没有限制。另外，对于1 <γ<3，我们为任意经典解的密度提供时间依赖性的下限，而没有对初始数据的任何其他假设。
In this paper, we consider the compressible Euler equations with time-dependent damping \frac{\a}{(1+t)^λ}u in one space dimension. By constructing 'decoupled' Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation of shock wave, if the derivatives of initial data are appropriately large at a point even when the damping coefficient goes to infinity with a algebraic growth rate. We study the case λ\neq1 and λ=1 respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for 1<γ<3 we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.
