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 exponentially fast.
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.
We considered classical solutions to the initial boundary value problem for non-isentropic compressible Euler equations with damping in multi-dimensions. We obtained global a priori estimates and global existence results of classical solutions to both non-isentropic Euler equations with damping and their nonlinear diffusion equations under small data assumption. We proved the pressure and velocity decay exponentially to constants, while the entropy and density can not approach constants. Finally, we proved the pressure and velocity of the non-isentropic Euler equations with damping converge ex…
Regularity and stability of two coupled Euler-Bernoulli equations with a localized singular structural damping
University of Monastir（莫纳斯提尔大学）；Higher Institute of Applied Mathematics and Computer Science, University of Kairouan（凯鲁万大学应用数学与计算机科学高等研究院）；Florida International University（佛罗里达国际大学）
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 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 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…
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…
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…
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…
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…
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…
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 will study the asymptotic behaviour of the energy decay of a transmission plate equation with locally distributed Kelvin-Voigt feedback. Precisly, we shall prove that the energy decay at least logarithmically over the time. The originality of this method comes from the fact that using a Carleman estimate for a transmission second order system which will be derived from the plate equation to establish a resolvent estimate which provide, by the famous Burq's result [Bur98], the kind of decay mentionned above.
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 …
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
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）
By rewriting a bipolar Euler-Poisson equations with damping into an Euler equation with damping coupled with an Euler-Poisson equation with damping, and using a new spectral analysis, we obtain the optimal decay results of the solutions in $L^2$-norm, which improve theose in \cite{Li3, Wu3}. More precisely, the velocities $u_1,u_2$ decay at the $L^2-$rate $(1+t)^{-{5}{4}}$, which is faster than the normal $L^2-$rate $(1+t)^{-{3}{4}}$ for the Heat equation and the Navier-Stokes equations. In addition, the disparity of two densities $ρ_1-ρ_2$ and the disparity of two velocities $u_1-u_2$ decay a…
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.
Well-Posedness and Asymptotic Decay of Solutions to the Three-Dimensional Euler Equations with Damping
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；School of Mathematics and Statistics, Anhui Normal University（安徽师范大学数学与统计学院）；School of Mathematical Sciences, Jiangsu University（江苏大学数学与统计学院）；College of Mathematics and Physics, Beijing University of Chemical Technology（北京化工大学理学院）
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.
We construct the global solution to the Cauchy's problem of the bipolar Euler-Poisson equations with damping in $\mathbb{R}^3$ when $H^3$ norm of the initial data is small. If further, the $\dot{H}^{-s}$ norm ($0\leq s<3/2)$ or $\dot{B}_{2,\infty}^{-s}$ norm ($0<s\leq3/2$) of the initial data is bounded, we give the optimal decay rates of the solution. As a byproduct, the decay results of the $L^p-L^2$ ($1\leq p\leq2$) type hold without the smallness of the $L^p$ norm of the initial data. In particular, we deduce that $\|\nabla^k(ρ_1-ρ_2)\|_{L^2} \sim(1+t)^{-5/4-\frac{k}{2}}$ and $\|\nabla^k(ρ…
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…
We consider admissible weak solutions to the compressible Euler system with source terms, which include rotating shallow water system and the Euler system with damping as special examples. In the case of anti-symmetric sources such as rotations, for general piecewise Lipschitz initial densities and some suitably constructed initial momentum, we obtain infinitely many global admissible weak solutions. Furthermore, we construct a class of finite-states admissible weak solutions to the Euler system with anti-symmetric sources. Under the additional smallness assumption on the initial densities, we…
