Asymptotic Behavior of Solutions to Euler Equations with Time-Dependent Damping in Critical Case

Source abstract

In this paper, we are concerned with the system of Euler equations with time-dependent damping like $-\frac{\mu}{(1+t)^\lambda}u$ for physical parameters $\lambda\ge 0$ and $\mu>0$. It is well known that, when $0\le \lambda<1$, the time-asymptotic-degenerate damping plays the key role which makes the damped Euler system behave like time-degenerate diffusion equations, while, when $\lambda>1$, the damping effect becomes really weak and can be neglected, which makes the dynamic system essentially behave like a hyperbolic system, and the singularity of solutions like shock waves will form. However, in the critical case $\lambda=1$, when $0<\mu\le 2$, the solutions of the system will blow up, but when $\mu>2$, the system is expected to possess global solutions. Here, we are particularly interested in the asymptotic behavior of the solutions in the critical case. By a heuristical analysis (variable scaling technique), we realize that, in this critical case, the hyperbolicity and the damping effect both play crucial roles and cannot be neglected. We first artfully construct the asymptotic profile, a special linear wave equation with time-dependent damping, which is totally different from the case of $0\leq\lambda<1,\mu>0$, whose profile is a self-similar solution to the corresponding parabolic equation. Then we rigorously prove that the solutions time-asymptotically converge to the solutions of linear wave equations with critical time-dependent damping. The convergence rates shown are optimal, by comparing with the linearized equations. The proof is based on the technical time-weighted energy method, where the time-weight is dependent on the parameter $\mu$.