Global existence and convergence to the modified Barenblatt solution for the compressible Euler equations with physical vacuum and time-dependent damping
Xinghong Pan
备注：
44 pages. All comments are welcome. arXiv admin note: substantial text overlap with arXiv:1407.6111 by other authors
发表信息：
Calc. Var. Partial Differential Equations 60 (2021)
In this paper, the smooth solution of the physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the vacuum boundary, the sound speed is $C^{1/2}$-Hölder continuous. The coefficient of the damping depends on time, given by this form $\fracμ{(1+t)^λ}$, $λ$, $μ>0$, which decays by order $-λ$ in time. Under the assumption that $0<λ<1$, $0<μ$ or $λ=1$, $2<μ$, we will prove the global existence of smooth solutions and convergence to the modified Barenblatt solution of the related porous media equation with time-dependent dissipation and the same total mass when the initial data of the Euler equations is a small perturbation of that of the Barenblatt solution. The pointwise convergence rates of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on space-time weighted energy estimates, elliptic estimates and Hardy inequality in the Lagrangian coordinates. Our result is an extension of that in Luo-Zeng [Comm. Pure Appl. Math. 69 (2016), no. 7, 1354-1396], where the authors considered the physical vacuum free boundary problem of the compressible Euler equations with constant-coefficient damping.
The large time behavior of entropy solution to the compressible Euler equations for polytropic gas (the pressure $p(ρ)=κρ^γ, γ>1$) with time dependent damping like $-\frac{1}{(1+t)^λ}ρu$ ($0<λ<1$) is investigated. By introducing an elaborate iterative method and using the intensive entropy analysis, it is proved that the $L^\infty$ entropy solution of compressible Euler equations with finite initial mass converges strongly in the natural $L^1$ topology to a fundamental solution of porous media equation (PME) with time-dependent diffusion, called by generalized Barenblatt solution. It is intere…
For the physical vacuum free boundary problem with the sound speed being $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the the porous media equation with the same total mass when the initial data is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in supereme norm and the precise expanding rate of the …
Global Existence and Asymptotic Equivalence to Barenblatt-type Solutions for the Physical Vacuum Free Boundary Problem of Damped Compressible Euler Equations in M-D
We consider the isothermal Euler system with damping. We rigorously show the convergence of Barenblatt solutions towards a limit Gaussian profile in the isothermal limit $γ$ $\rightarrow$ 1, and we explicitly compute the propagation and the behavior of Gaussian initial data. We then show the weak L 1 convergence of the density as well as the asymptotic behavior of its first and second moments. Contents 1. Introduction 1 2. Assumptions and main results 3 3. The limit $γ$ $\rightarrow$ 1 of Barenblatt's solutions 6 4. Gaussian solutions 9 5. Evolution of certain quantities 10 6. Convergence 15 7…
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The global existence of smooth solutions to the vacuum free boundary problem with physical singularity of compressible Euler equations with damping and gravity is proved in space dimensions $n=1, 2, 3$, for the initial data being small perturbations of the stationary solution. Moreover, the exponential decay of the velocity is obtained for $n=1, 2, 3$. The exponentially fast convergence of the density and vacuum boundary to those of the stationary solution is shown for $n=1$, and it is proved for $n=2, 3$ that they stay close to those of the stationary solution if they do so initially. The pro…
We study the exponential stability of constant steady state of isentropic compressible Euler equation with damping on $\mathbb T^n$. The local existence of solutions is based on semigroup theory and some commutator estimates. We propose a new method instead of energy estimates to study the stability, which works equally well for any spatial dimensions.
In this paper, we are concerned with the global existence and blowup of smooth solutions of the 3-D compressible Euler equation with time-depending damping $$ \partial_tρ+\operatorname{div}(ρu)=0, \quad \partial_t(ρu)+\operatorname{div}\left(ρu\otimes u+p\,I_{3}\right)=-\,\fracμ{(1+t)^λ}\,ρu, \quad ρ(0,x)=\bar ρ+\varepsilonρ_0(x),\quad u(0,x)=\varepsilon u_0(x), $$ where $x\in\mathbb R^3$, $μ>0$, $λ\geq 0$, and $\barρ>0$ are constants, $ρ_0,\, u_0\in C_0^{\infty}(\mathbb R^3)$, $(ρ_0, u_0)\not\equiv 0$, $ρ(0,\cdot)>0$, and $\varepsilon>0$ is sufficiently small. For $0\leqλ\leq1$, we show that …
Stochastic Compressible Euler Equations with Frictional Damping: Existence of $L^\infty$ Martingale Solutions and Asymptotic Porous Medium-Like Behavior
University of California Berkeley（加州大学伯克利分校）；University of Maryland（马里兰大学）；University of Washington（华盛顿大学）
We study the compressible quantum Navier-Stokes (QNS) equations with degenerate viscosity in the three dimensional periodic domains. On the one hand, we consider QNS with additional damping terms. Motivated by the recent works [Li-Xin, arXiv:1504.06826] and [Antonelli-Spirito, Arch. Ration. Mech. Anal., 203(2012), 499--527], we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the BD entropy estimate. Using this system, we obtain the global existence of weak solutions to the compressible QNS equations with damping terms for large initial da…
Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping
School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）；Department of Mathematics, University of Notre Dame（圣母大学数学系）；School of Mathematics and Statistics, Guangdong University of Technology（广东工业大学数学与统计学院）
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