Well-posedness of the IBVP for 2-D Euler Equations with Damping
Yongqin Liu, Weike Wang
本文研究了一类带阻尼项的二维等熵Euler方程的初边值问题。利用能量估计的方法证明了该初边值问题经典解的整体存在性。
In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial-boundary value problem by the method of energy estimates.
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In [2] we introduced a method combining together an observability inequality and a spectral decomposition to get a logarithmic stability estimate for the inverse problem of determining both the potential and the damping coefficient in a dissipative wave equation from boundary measurements. The present work deals with an adaptation of that method to obtain a logarithmic stability estimate for the inverse problem of determining a boundary damping coefficient from boundary measurements. As in our preceding work, the different boundary measurements are generated by varying one of the initial condi…
Optimal decay rate of the bipolar Euler-Poisson system with damping in $\mathbb{R}^3$
通过将一个双极阻尼Euler- Poisson方程改写为一个阻尼Euler方程与一个阻尼Euler- Poisson方程的耦合，利用一种新的谱分析方法，得到了解在$L^2$范数下的最优衰减结果，改进了\cite{Li3, Wu3}中的结果.更准确地说，速度$u_1,u_2$以$L^2-$速率$(1+t)^{-{5}{4}}$衰减，该速率比Heat方程和Navier-Stokes方程的正常$L^2-$速率$(1+t)^{-{3}{4}}$更快。另外，两个密度的差异$ρ_1-ρ_2$和两个速度的差异$u_1-u_2$以$L^2$ -速率$(1+t)^{-2}$衰减。
We find a sharp condition on the density-dependent coefficient of damping of a one-dimensional repulsive Euler-Poisson system, which makes it possible to suppress the formation of singularities in the solution of the Cauchy problem with arbitrary smooth data. In the context of plasma physics, this means the possibility of suppressing the breakdown of arbitrary oscillations of cold plasma.
Euler-Poisson equations with velocity-dependent damping
研究一维及多维径向对称的排斥性欧拉-泊松方程，假设幂律速度依赖阻尼系数，发现一维中对应全局光滑解的初值集扩展，其他维度（除4维）阻尼不改善解的光滑性，稳态小扰动会爆破。
We revisit a family of infinite-energy solutions of the 3D incompressible Euler equations proposed by Gibbon et al. [9] and shown to blowup in finite time by Constantin [6]. By adding a damping term to the momentum equation we examine how the damping coefficient can arrest this blowup. Further, we show that similar infinite-energy solutions of the inviscid 3D Boussinesq system with damping can develop a singularity in finite time as long as the damping effects are insufficient to arrest the (undamped) 3D Euler blowup in the associated damped 3D Euler system.
In mathematical physics, the pressure function is determined by the equation of state. There are two existing barotropic state equations: the state equation for polytropic gas with adiabatic index greater than or equal to 1 and the state equation for generalized Chaplygin gas in cosmology. In this paper, a logarithmic pressure is derived naturally with the coexistence of the two existing state equations through an equivalent symmetric hyperbolic transformation. On the study of the logarithmic pressure, global existence of solutions with small initial data to the one-dimensional compressible Eu…
We study the asymptotic behavior of the forced linear Euler and nonlinear Navier-Stokes equations close to Couette flow in a periodic channel. As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $H^{s}, s>-1$. We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier-Stokes equations. Hence, these results provide an indication that nonli…
Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques
本文引入Orlicz空间理论工具处理压强非线性项，给出了带时变阻尼的可压缩Euler系统在n=2、3维下仅依赖维度和阻尼强度的寿命统一上界，改进了已有结果且具备相关研究应用潜力。
Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations
在$\Torus \times \Real$上证明了二维无粘Euler方程中剪切流接近于平面Couette流的渐近稳定性。也就是说，如果库埃特流的初始扰动在适当的正则类中很小，特别是在小于2类的Gevrey空间中，速度在L^2中强烈地收敛到剪切流，该剪切流也接近库埃特流。涡度由线性演化渐近驱动到小尺度，并弱收敛为$t \rightarrow \pm\infty$。速度场的强收敛有时被称为无粘阻尼，这是由于与Vlasov方程中的朗道阻尼的关系。这种收敛是正式推导出在线性水平的开尔文在1887年和它发生在代数率首先计算奥尔在1907年;我们的工作似乎是第一次严格确认这种行为的非线性水平。
Landau damping: paraproducts and Gevrey regularity
我们给出了一个新的、更简单的证明，证明了T^d上的非线性朗道阻尼满足Gevrey-1/s正则性（s > 1/3），这与Mouhot和Villani在朗道阻尼的原始证明[Acta Mathematica 2011]中的形式分析所预测的正则性要求相匹配。我们的证明以一种新颖的方式结合了Landau阻尼的原始证明和2D Euler中无粘阻尼的证明[arXiv：1306.5028]。在2D欧拉的工作中，我们使用paraproduct分解和控制正则性损失，以取代牛顿迭代方案中采用的原始证明。我们执行的时间响应估计从原来的证明，以控制等离子体回波，并将它们耦合到能量估计的分布函数的风格的工作在2D欧拉。
We consider various questions about the 2d incompressible Navier-Stokes and Euler equations on a torus when dissipation is removed from or added to some of the Fourier modes.
We study a Euler-Bernoulli beam equation with localized discontinuous structural damping. As our main result, we prove that the associated $C_0$-semigroup $(S(t))_{t\geq0}$ is of Gevrey class $δ>24$ for $t>0$, hence immediately differentiable. Moreover, we show that $(S(t))_{t\geq0}$ is exponentially stable.
Global Existence and Nonlinear Diffusion of Classical Solutions to Non-Isentropic Euler Equations with Damping in Bounded Domain
研究了多维带阻尼非等熵可压缩Euler方程初边值问题的经典解。在小数据假设下，得到了非等熵阻尼Euler方程及其非线性扩散方程的经典解的整体先验估计和整体存在性结果.我们证明了压力和速度指数衰减到常数，而熵和密度不能接近常数。最后证明了当时间趋于无穷大时，带阻尼的非等熵Euler方程的压力和速度指数收敛于其非线性扩散方程的压力和速度.
Zero-relaxation and vanishing-damping limits of pressureless Euler system
研究一维无压力Euler系统在Radon测度空间中的零松弛极限（收敛到静态解）和消失阻尼极限（收敛到无阻尼系统解），揭示松弛与阻尼对解行为的影响。
Regularity and stability of two coupled Euler-Bernoulli equations with a localized singular structural damping
本文研究了带有局部奇异阻尼的双耦合欧拉-伯努利方程组的正则性与稳定性，推导了均匀稳定性结果并描述了系统长期行为。
Blowup of Solutions to a Damped Euler Equation with Homogeneous Three-Point Boundary Condition
本文研究带阻尼的广义无粘Proudman-Johnson方程在齐次三点边界条件下解的有限时间爆破，推导了初始数据需满足的条件，适用于有界或无界光滑阻尼项。
We study the Cauchy problem for a system of equations corresponding to a singular limit of radiative hydrodynamics, namely the 3D radiative compressible Euler system coupled to an electromagnetic field through the MHD approximation. Assuming the presence of damping together with suitable smallness hypotheses for the data, we prove that this problem admits a unique global smooth solution.
In this paper, vacuum and singularity formation are considered for compressible Euler equations with time-dependent damping. For $1<γ\leq 3$, by constructing some new control functions ingeniously, we obtain the lower bounds estimates on density for arbitrary classical solutions. Basing on these lower estimates, we succeed in proving the singular formation theorem for all $λ$, which was open in [1] for some cases.Moreover, the singularity formation of the compressible Euler equations when $γ=3$ is investigated, too.
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.
We consider the Euler-Bernoulli beam equation with a local Kelvin-Voigt dissipation type in the interval $(-1,1)$. The coefficient damping is only effective in $(0,1)$ and is degenerating near the $0$ point with a speed at least equal to $x^α$ where $α\in(0,5)$. We prove that the semigroup corresponding to the system is polynomially stable and the decay rate depends on the degeneracy speed $α$.
In 1992, Hsiao and Liu \cite{Hsiao-Liu-1} firstly showed that the solution to the compressible Euler equations with damping time-asymptotically converges to the diffusion wave $(\bar v, \bar u)$ of the porous media equation. In \cite{Geng-Huang-Jin-Wu}, we proposed a time-asymptotic expansion around the diffusion wave $(\bar v, \bar u)$, which is a better asymptotic profile than $(\bar v, \bar u)$. In this paper, we rigorously justify the time-asymptotic expansion by the approximate Green function method and the energy estimates. Moreover, the large time behavior of the solution to compressibl…
We study the isentropic Euler equations with time-dependent damping, given by $\fracμ{(1+t)^λ}ρu$. Here, $λ,μ$ are two non-negative constants to describe the decay rate of damping with respect to time. We will investigate the global existence and asymptotic behavior of small data solutions to the Euler equations when $0<λ<1,0<μ$ in multi-dimensions $n\geq 1$. The asymptotic behavior will coincide with the one that obtained by many authors in the case $λ=0$. We will also show that the solution can only decay polynomially in time while in the three dimensions, the vorticity will decay exponentia…
The time asymptotic expansion for the compressible Euler equations with time-dependent damping
本文针对含时变阻尼的可压缩欧拉方程组，提出围绕广义多孔介质方程自相似解的时间渐近展开，并严格证明其在λ∈(1/7,1)时为最佳渐近轮廓。
We study the 1-d isentropic Euler equations with time-decayed damping \begin{equation} \left\{ \begin{aligned} &\partial_t ρ+\partial_x(ρu)=0, \\ &\partial_t(ρu)+ \partial_x(ρu^2)+\partial_xp(ρ)=-\fracμ{1+t}ρu,\\ &ρ|_{t=0}=1+\varepsilonρ_0(x),u|_{t=0}=\varepsilon u_0(x). \end{aligned} \right. \nonumber \end{equation} This work is inspired by a recent work of F. Hou, I. Witt and H.C. Yin \cite{Hou01}. In \cite{Hou01}, they proved a global existence and blow-up result of 3-d irrotational Euler flow with time-dependent damping. In the 1-d case, we will prove a different result when the damping de…
The global well-posedness of the multi-dimensional compressible Euler system with damping in the $L^p$ critical Besov spaces for $p<2$
本文在$L^p$临界贝塞尔空间中研究了具有阻尼的多维可压缩欧拉系统的全局适定性，通过建立新的乘积估计来证明结果。
Shifted shock formation for the 3D compressible Euler equations with damping and variation of the vorticity
本文研究带阻尼的3D可压缩Euler方程的激波形成问题，证明阻尼无法阻止点激波形成但会移动激波时间与波幅，且涡量在非爆破方向呈指数变化。
This paper is concerned with the multi-dimensional compressible Euler equations with time-dependent damping of the form $-\fracμ{(1+t)^λ}ρ\boldsymbol u$ in $\mathbb R^n$, where $n\ge2$, $μ>0$, and $λ\in[0,1)$. When $λ>0$ is bigger, the damping effect time-asymptotically gets weaker, which is called under-damping. We show the optimal decay estimates of the solutions such that $\|\partial_x^α(ρ-1)\|_{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+λ}{2}(\frac{n}{2}+|α|)}$, and $\|\partial_x^α\boldsymbol u\|_{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+λ}{2}(\frac{n}{2}+|α|)-\frac{1-λ}{2}}$, and see how th…
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…
In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping for the occurrence of the finite time blow-up. In particular, our sufficient conditions ensure that the derivative blow-up occurs in finite time with the solution itself and the pressure bounded. Our method is based on simple estimates with Riemann invariants. Furthermore, we give sharp lower and upper estimates of the lifespan of solutions, when initial dat…
Existence and asymptotic behavior of $C^1$ solutions to the multidimensional compressible Euler equations with damping
本文在Besov空间框架下研究了具有阻尼项的多维可压缩Euler方程的$C^1$解的存在性和渐近性态。我们弱化了对初始数据的正则性要求，改进了Sideris-P.D.E. -Wang（Comm.P.D.E. 28（2003）953）。全局解的存在性依赖于一个关键的先验估计，并通过谱局部化方法证明了该先验估计的存在性。主要的分析工具是Littlewood-Paley分解和Bony的仿积公式。
In this paper, we show the shock formation to the compressible Euler equations with time-dependent damping $\frac{a\p u}{(1+t)^{\lam}}$ in three spatial dimensions without any symmetry conditions. It's well-known that for $\lam>1$, the damping is too weak to prevent the shock formation for suitably large data. However, the classical results only showed the finite existence of the solution. Follow the work by D.Christodoulou in\cite{christodoulou2007}, starting from the initial isentropic and irrotational short pulse data, we show the formation of shock is characterized by the collapse of the c…
In this paper, we consider the one-dimensional isentropic compressible Euler equations with linear damping $β(t,x)ρu$ in a bounded domain, which can be used to describe the process of compressible flows through a porous medium.~And the model is imposed a dissipative subsonic time-periodic boundary condition.~Our main results reveal that the time-periodic boundary can trigger a unique subsonic time-periodic smooth solution which is stable under small perturbations on initial data. Moreover, the time-periodic solution possesses higher regularity and stability provided a higher regular boundary c…
In this paper, we consider the 1D compressible Euler equation with the damping coefficient $λ/(1+t)^μ$. Under the assumption that $0\leq μ<1$ and $λ>0$ or $μ=1$ and $λ> 2$, we prove that solutions exist globally in time, if initial data are small $C^1$ perturbation near constant states. In particular, we remove the conditions on the limit $\lim_{|x| \rightarrow \infty} (u (0,x), v (0,x))$, assumed in previous results.
In this note we revisit the proof of Bedrossian and Masmoudi [arXiv:1306.5028] about the inviscid damping of planar shear flows in the 2D Euler equations under the assumption of zero mean perturbation. We prove that a small perturbation to the 2D Euler Couette flow in $\mathbb{T}\times \mathbb{R}$ strongly converge to zero, under the additional assumption that the average in $x$ is always zero. In general the mean is not a conserved quantity for the nonlinear dynamics, for this reason this is a particular case. Nevertheless our assumption allow the presence of echoes in the problem, which we c…
In this paper, we study the relaxation limit of the relaxing Cauchy problem for non-isentropic compressible Euler equations with damping in multi-dimensions. We prove that the velocity of the relaxing equations converges weakly to that of the relaxed equations, while other variables of the relaxing equations converges strongly to the corresponding variables of the relaxed equations. We show that as relaxation time approaches 0, there exists an initial layer for the ill-prepared data, the convergence of the velocity is strong outside the layer; while there is no initial layer for the well-prepa…
Sharp lifespan estimate for the compressible Euler system with critical time-dependent damping in $\R^2$
本文研究了 $\R^2$ 中带临界时间依赖阻尼的可压缩 Euler 系统光滑解的长时间存在性，通过引入向量场 $\widehat{Z}$ 替代传统向量场 $Z$ 以改善线性误差项衰减，建立了关于初始扰动小参数的下界精确寿命估计，该思想亦适用于非线性波动方程的长时间行为研究。
Large-time behavior of solutions to bipolar Euler-Poisson equations with space-dependent damping
本文通过加权能量方法证明，在空间相关阻尼下，一维双极Euler-Poisson方程Cauchy问题的小扰动初值存在唯一全局光滑解，并以与常数阻尼相同的代数速率收敛到稳态，且无需阻尼系数的小性条件。
In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler- Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on t…
We prove the asymptotic stability of shear flows close to the Couette flow for the 2-D inhomogeneous incompressible Euler equations on $\mathbb{T}\times \mathbb{R}$. More precisely, if the initial velocity is close to the Couette flow and the initial density is close to a positive constant in the Gevrey class 2, then 2-D inhomogeneous incompressible Euler equations are globally well-posed and the velocity converges strongly to a shear flow close to the Couette flow, and the vorticity will be driven to small scales by a linear evolution and weakly converges as $t\to \infty$. To our knowledge, t…
We study the asymptotic behaviour for a system consisting of a clamped flexible beam that carries a tip payload, which is attached to a nonlinear damper and a nonlinear spring at its end. Characterizing the omega-limit sets of the trajectories, we give a necessary condition under which the system is asymptotically stable. In the case when this condition is not satisfied, we show that the beam deflection approaches a non-decaying time-periodic solution.
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…
This paper is concerned with the multi-dimensional compressible Euler equations with time-dependent over-damping of the form $-\fracμ{(1+t)^λ}ρ\boldsymbol u$ in $\mathbb R^n$, where $n\ge2$, $μ>0$, and $λ\in[-1,0)$. This continues our previous work dealing with the under-damping case for $λ\in[0,1)$. We show the optimal decay estimates of the solutions such that for $λ\in(-1,0)$ and $n\ge2$, $\|ρ-1\|_{L^2(\mathbb R^n)}\approx(1+t)^{-\frac{1+λ}{4}n}$ and $\|\boldsymbol u\|_{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+λ}{4}n-\frac{1-λ}{2}}$, which indicates that a stronger damping gives rise to sol…
In this paper, we show the shock formation of the solutions to the 3-dimensional (3D) compressible isentropic and irrotational Euler equations with damping for the initial short pulse data which was first introduced by D.Christodoulou\cite{christodoulou2007}. Due to the damping effect, the largeness of the initial data is necessary for the shock formation and we will work on the class of large data (in energy sense). Similar to the undamped case, the formation of shock is characterized by the collapse of the characteristic hypersurfaces and the vanishing of the inverse foliation density functi…
Stabilities for Euler-Poisson Equations in Some Special Dimensions
研究了天体物理中描述气态恒星演化的欧拉-泊松方程的稳定性和经典解。实际上，我们将带摩擦阻尼项和不带摩擦阻尼项的Euler-泊松方程的稳定性研究推广到了一些特殊的维空间。此外，利用二维Euler-泊松方程的第二惯性函数，证明了对任意$γ$，具有$2\int_Ω(ρ| u| ^{2}+2P)dx<gM^{2}-ε$的经典解的非整体存在性.
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 respecti…
Time periodic problem of compressible Euler equations with damping on the whole space
本文研究了全空间上带阻尼可压缩Euler方程的时间周期问题，通过位势理论估计和带权能量估计，在空间维数大于等于3时证明了充分小外力下时间周期解的存在性与渐近稳定性。
Explicit solutions and linear inviscid damping in the Euler-Boussinesq equation near a stratified Couette flow in the periodic strip
本文针对周期条带中稳定分层Couette流附近的线性化Boussinesq方程，构造显式解，并利用极限吸收原理与振荡积分方法证明任意正Richardson数下扰动密度和速度场的最优无粘阻尼。
Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non-constant density in a finite channel
本文证明了在有限通道内，具有非恒定背景密度的二维非齐次欧拉方程中，一类单调剪切流在 Gevrey 类紧支初值扰动下具有非线性无粘阻尼。
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 dissipat…
This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant $(x,t)\in \mathbb{R}^+\times\mathbb{R}^+$, \begin{equation}\notag \partial_t v - \partial_x u=0, \qquad \partial_t u + \partial_x p(v) =\displaystyle -\fracα{(1+t)^λ} u, \end{equation} with null-Dirichlet boundary condition or null-Neumann boundary condition on $u$. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends time-asymptotically to the nonlinear diffusion wave. Compared with the previous work abo…
Logarithmic stabilization of the Euler-Bernoulli transmission plate equation with locally distributed Kelvin-Voigt damping
本文研究具有局部分布Kelvin-Voigt反馈的透射板方程能量衰减的渐近性态。确切地说，我们将证明能量至少随时间以几何学的方式衰减。这种方法的独创性来自于这样一个事实，即使用Carleman估计的传输二阶系统，这将是来自板方程建立一个预解估计，提供了著名的Burq的结果[Bur 98]，衰减上述类型。
We prove a definitive theorem on the asymptotic stability of point vortex solutions to the full Euler equation in 2 dimensions. More precisely, we show that a small, Gevrey smooth, and compactly supported perturbation of a point vortex leads to a global solution of the Euler equation in 2D, which converges weakly as $t\to\infty$ to a radial profile with respect to the vortex. The position of the point vortex, which is time dependent, stabilizes rapidly and becomes the center of the final, radial profile. The mechanism that leads to stabilization is mixing and inviscid damping.
Optimal time-decay for Euler-Fourier system with damping in the critical $L^2$ framework
本文在临界 $L^2$ 框架下，通过时间加权能量方法和 Lyapunov 泛函分析，去除了初始数据低频小性条件，建立了带阻尼 Euler-Fourier 系统解的最优时间衰减率。
On Vacuum Free Boundary Problem of the Spherically Symmetric Euler Equations with Damping and Solid Core
本文研究带阻尼与固体核心的球对称欧拉方程真空自由边界问题，在γ>4/3和小初值条件下证明光滑解全局存在及平衡态指数时间衰减稳定。
Global dynamics of damped Euler systems with exterior potentials
本文研究带线性阻尼与外势的三维等温 Euler 方程，分别证明大阻尼大数据全局适定性、小数据经典解的最优代数衰减率，以及无压情形阻尼不足时的有限时间爆破。
Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping
对于声速为$C^{{1}/{2}}$ lder的一维可压缩阻尼Euler方程的近真空边界连续的物理真空自由边界问题，证明了光滑解的整体存在性，它被证明是收敛到巴伦布拉特自我，当初始数据是Barenblatt解的一个小扰动时，对于总质量相同的多孔介质方程，得到了类似的解。给出了密度的逐点收敛速度、速度的超实范数收敛速度和真空边界的精确扩张速度。证明是基于一个建设的高阶加权泛函的空间和时间的权重捕获的行为的解决方案在真空状态附近和大的时间，一个新的分析，高阶非线性能量估计和椭圆估计的介绍。
Compressible Euler equations with time-dependent damping in the critical regularity setting: global well-posedness and strong relaxation limit
本文研究了具有时间依赖阻尼的压缩欧拉方程在临界正则性下的全局well-posed性及强松弛极限，通过改进的超协调框架和时间依赖频域分解，证明了经典解的全局存在性和强收敛性。
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 …
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…
Stochastic Compressible Euler Equations with Frictional Damping: Existence of $L^\infty$ Martingale Solutions and Asymptotic Porous Medium-Like Behavior
本文研究了一维等熵可压缩欧拉方程在线性摩擦阻尼作用下的随机行为，证明了$L^\infty$马尔可夫解的存在性，并展示了在合适噪声条件下解几乎必然且指数收敛到常稳态，其密度近似由确定性多孔介质方程的渐近解描述。
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