We prove that the smooth solutions to the Cauchy problem for the three-dimensional compressible barotropic magnetohydrodynamic equations with conserved total mass and finite total energy lose the initial smoothness within a finite time. Further, we show that the same result holds for the solution to the Cauchy problem for the multidimensional compressible Navier-Stokes system. Moreover, for the solution with a finite momentum of inertia we get the two-sided estimates of different components of total energy.
We study the lagrangian structure for weak solutions of two dimensional Navier-Stokes equations for a non-barotropic compressible fluid, i.e. we show the uniqueness of particle trajectories for two dimensional compressible fluids including the energy equation. Our result extends partially the result obtained for barotropic fluids by Hoff and Santos (2008).
We study the system of equations which describes barotropic (isentropic) flows of viscous compressible multi-fluids (mixtures of fluids). We study the relations between pressure, densities, concentrations, viscosities and other parameters of the flow and of the medium.
We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscousity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper.
We consider the model of viscous compressible homogeneous multi-fluids with multiple velocities. We review different formulations of the model and the existence results for boundary value problems. We analyze crucial mathematical difficulties which arise during the proof of the existence theorems.
We consider the equations of a multi-velocity model of a binary mixture of viscous compressible fluids (two-fluid medium) in the case of one-dimensional barotropic motions. We prove the global (in time) existence and uniqueness of a strong solution to the initial-boundary value problem describing the motion in a bounded spatial domain.
We study the long-time behavior of a point mass moving in a one-dimensional viscous compressible fluid. Previously, we showed that the velocity of the point mass $V(t)$ satisfies a decay estimate $V(t)=O(t^{-3/2})$~[K. Koike, J. Differential Equations \textbf{271} (2021) 356--413]. This result was obtained as a corollary to pointwise estimates of solutions to a free boundary problem of barotropic compressible Navier--Stokes equations. In this paper, we give a simple necessary and sufficient condition on the initial data for the decay estimate $V(t)=O(t^{-3/2})$ to be optimal. This is achieved …
In this paper we consider the barotropic compressible quantum Navier-Stokes equations with a linear density dependent viscosity and its limit when the scaled Planck constant vanish. Following recent works on degenerate compressible Navier-Stokes equations, we prove the global existence of weak solutions by the use of a singular pressure close to vacuum. With such singular pressure, we can use the standard definition of global weak solutions which also allows to justify the limit when the scaled Planck constant denoted by $ε$ tends to 0.
This paper concerns the existence of global weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients. We construct suitable approximate system which has smooth solutions satisfying the energy inequality, the BD entropy one, and the Mellet-Vasseur type estimate. Then, after adapting the compactness results due to Mellet-Vasseur [Comm. Partial Differential Equations 32 (2007)], we obtain the global existence of weak solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in two or three dimensiona…
In this paper, we study some model problem associated to the free boundary value problem of the barotropic compressible Navier-Stokes equations in general smooth domain with taking surface tension into account. To obtain the maximal $L_p-L_q$ regularity property of the model problem, we prove the existence of $\mathcal{R}-$bounded operator families of the resolvent problem via Weis' theory on operator valued Fourier multipliers.
For barotropic compressible Navier-Stokes equations in three-dimensional (3D) bounded domains, we prove that any finite energy weak solution obtained by Lions [Mathematical topics in fluid mechanics, Vol. 2. Compressible models(1998)] and Feireisl-Novotný-Petzeltová [J. Math. Fluid Mech. 3(2001), 358-392] decays exponentially to the equilibrium state. This result is established by both using the extra integrability of the density due to Lions and constructing a suitable Lyapunov functional just under the framework of Lions-Feireisl's weak solutions.
This paper concerns the Cauchy problem of the barotropic compressible Navier-Stokes equations on the whole two-dimensional space with vacuum as far field density. In particular, the initial density can have compact support. When the shear and the bulk viscosities are a positive constant and a power function of the density respectively, it is proved that the two-dimensional Cauchy problem of the compressible Navier-Stokes equations admits a unique local strong solution provided the initial density decays not too slow at infinity. Moreover, if the initial data satisfy some additional regularity …
The aim of this work is to prove the weak-strong uniqueness principle for the compressible Navier-Stokes-Poisson system on an exterior domain, with an isentropic pressure of the type $p(\varrho)=a\varrho^γ$ and allowing the density to be close or equal to zero. In particular, the result will be first obtained for an adiabatic exponent $γ\in [9/5,2]$ and afterwards, this range will be slightly enlarged via pressure estimates "up to the boundary", deduced relaying on boundedness of a proper singular integral operator.
We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially. We first prove the local existence and uniqueness of the strong solutions, where the initial compatibility condition proposed in \cite{cho1,K2,coi1} is removed under suitable sense. Then, the continuous of strong solutions on the initial data is derived under an additional compatibility condition. Moreover, for the initial data satisfying some additional regularity and compatibility condition, the strong soluti…
In this article, we study the blowup phenomena of compressible Euler equations with non-vacuum initial data. Our new results, which cover a general class of testing functions, present new initial value blowup conditions. The corresponding blowup results of the 1-dimensional case in non-radial symmetry are also included.
We study an initial boundary value problem for the Navier-Stokes equations of compressible viscous heat-conductive fluids in a 2-D periodic domain or the unit square domain. We establish a blow-up criterion for the local strong solutions in terms of the gradient of the velocity only, which coincides with the famous Beale-Kato-Majda criterion for ideal incompressible flows.
In this paper, we obtain a blow up criterion for strong solutions to the 3-D compressible Naveri-Stokes equations just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible flow. The key ingredients in our analysis are the a priori super-norm estimate of the momentum by a Moser-iteration and an estimate of the space-time square mean of the gradient of the density. In addition, initial vacuum is allowed in our case.
Initial-boundary value problem for linearized equations of motion of viscous barotropic fluid in a bounded domain is considered. Existence, uniqueness and estimates of weak solutions to this problem are derived. Convergence of the solutions towards the incompressible limit when compressibility tends to zero is studied.
We consider the barotropic Navier-Stokes system describing the motion of a compressible viscous fluid confined to a bounded domain driven by time periodic inflow/outflow boundary conditions. We show that the problem admits a time periodic solution in the class of weak solutions satisfying the energy inequality.
Low Mach number limit of a pressure correction MAC scheme for compressible barotropic flows
Aix-Marseille Université, CNRS, École Centrale de Marseille（艾克斯-马赛大学，法国国家科学研究中心，马赛中央理工学院）；Institut de Radioprotection et de Sûreté Nucléaire (IRSN)（核辐射防护与安全研究所）；Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan（里昂大学，法国国家科学研究中心UMR 5208，里昂第一大学，卡米尔·若尔当研究所）
We prove that the smooth solutions to the Cauchy problem for the Navier-Stokes equations with conserved mass, total energy and finite momentum of inertia loses the initial smoothness within a finite time in the case of space of dimension 3 or greater even if the initial data are not compactly supported.
We consider barotropic motions described by the compressible Navier-Stokes equations in a box with periodic boundary conditions. We are looking for density $\varrho$ in the form $\varrho=a+η$, where $a$ is a constant and $η|_{t=0}$ is sufficiently small in $H^2$-norm. We assume existence of potentials $φ$ and $ψ$ such that $v=\nablaφ+\mathrm{rot}ψ$. Next we assume that $\nablaφ|_{t=0}$ is sufficiently small in $H^2$-norm too. Finally, we assume that the second viscosity coefficient $ν$ is sufficiently large. Then we prove long time existence of solutions such that $v\in L_\infty(0,T;H^2(Ω))\ca…
This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies the Serrin's condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanish or…
In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure--valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong conver…
We consider viscous compressible barotropic motions in a bounded domain $Ω\subset \mathbb{R}^3$ with the Dirichlet boundary conditions for velocity. We assume the existence of some special sufficiently regular solutions $v_s$ (velocity), $\varrho_s$ (density) of the problem. By the special solutions we can choose spherically symmetric solutions. Let $v$, $\varrho$ be a~solution to our problem. Then we are looking for differences $u=v-v_s$, $η=\varrho-\varrho_s$. We prove existence of $u$, $η$ such that $u,η\in L_\infty(kT,(k+1)T;H^2(Ω))$, $u_t,η_t\in L_\infty(kT,(k+1)T;H^1(Ω))$, $u\in L_2(kT,(…
In this paper, we consider the initial-boundary value problem to the compressible Navier-Stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in $\mathbb R^N$, with $N\geq1$. We prove that any classical solutions $(ρ, u, θ)$, in the class $C^1([0,T]; H^m(Ω))$, $m>[\frac N2]+2$, with bounded from below initial entropy and compactly supported initial density, which allows to touch the physical boundary, must blow-up in finite time, as long as the initial mass is positive. This paper extends the classical reault by Xin [CPAM, 1998], in which the Cauch…
We prove the global existence and uniqueness of strong solutions for a compressible multifluid described by the barotropic Navier-Stokes equations in dim = 1. The result holds when the diffusion coefficient depends on the pressure. It relies on a global control in time of the L2 norm of the space derivative of the density, via a new kind of entropy.
Regularity and uniqueness of weak solutions of the compressible barotropic Navier-Stokes equations with constant viscosity coefficients is proven for small time in dimension $N=2,3$ under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to $P(ρ)=aρ^γ$ with $γ>1$ (in dimension three, additional conditions of size will be ask on $γ$). The second part of the paper is devoted to blow-up criteria for slightly subcritical initial data for the scaling of the equations whe…
Nonlinearly Exponential Stability for Lions-Feireisl's Weak Solutions to the Barotropic Compressible Navier-Stokes Equations with Large Potential External Forces
We study the construction of analytical non-radially solutions for the 1-dimensional compressible adiabatic Euler equations in this article. We could design the perturbational method to construct a new class of analytical solutions. In details, we perturb the linear velocity:% \begin{equation} u=c(t)x+b(t) \end{equation} and substitute it into the compressible Euler equations. By comparing the coefficients of the polynomial, we could deduce the corresponding functional differential system of $(c(t),b(t),ρ^{γ-1}(0,t)).$ Then by skillfully applying the Hubble's transformation: \begin{equation} c…
We establish a blow-up criterion in terms of the upper bound of the density and temperature for the strong solution to 2D compressible viscous heat-conductive flows. The initial vacuum is allowed.
We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for barotropic compressible fluids in $\mathbb{R}^3$. When the viscosity coefficients obey a lower power-law of the density (i.e., $ρ^δ$ with $0<δ<1$), we identify a quasi-symmetric hyperbolic--singular elliptic coupled structure of the Navier-Stokes equations to control the behavior of the velocity of the fluids near the vacuum. Then this structure is employed to prove that there exists a unique regular solution to the corresponding Cauchy problem with arbitrarily large initial data and far-field va…
We study a possibility of existence of localized two-dimensional structures, both smooth and non-smooth, that can move without significant change of their shape in a leading stream of compressible barotropic fluid on a rotating plane.
We consider the compressible (barotropic) Navier-Stokes system on time-dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behaviour, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained.
In this paper, we establish the existence and uniqueness of local strong solutions to the kinetic Cucker--Smale model coupled with the isentropic compressible Navier--Stokes equation in the whole space. Moreover, the blowup mechanism for strong solutions to the coupled system is also investigated.
In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data. In particular, the results hold true for the full compressible Euler equations and isentropic compressible Euler equations and the blow-up time can be computed in a more precise way. It is not required that the initial data has compact support or contain vacuum in any finite regions. Moreover, a simplifie…
We shall deal with both the barotropic and the full compressible Euler system in multiple space dimensions. Both systems are particular examples of hyperbolic conservation laws. Whereas for scalar conservation laws there exists a well-known complete well-posedness theory, and for one-dimensional systems one also has achieved several results on existence and uniqueness, in the case of multi-dimensional systems there are even negative results regarding uniqueness: With the so-called convex integration method it is possible to show that there exist initial data for which the compressible Euler eq…
In this paper, we shall prove a blow-up criteria in terms of the upper bound of the density, deformation tensor and the gradient of deformation tensor for 3D compressible visco-elasticity. Based on the structure of deformation tensor, we find a cancelation in higher order derivatives, which brings us the desired results.
We study the strong solution to the 3-D compressible Navier--Stokes equations. We propose a new blow up criterion for barotropic gases in terms of the integral norm of density $ρ$ and the divergence of the velocity $\bu$ without any restriction on the physical viscosity constants. Our blow up criteria can be seen as a partial realization of the underlying principle that the higher integrability implies the boundedness and then eventual regularity. We also present similar blow up criterion for the heat conducting gases.
We consider the barotropic Navier--Stokes system describing the motion of a viscous compressible fluid interacting with the outer world through general in/out flux boundary conditions. We consider a hard--sphere type pressure EOS and show that all trajectories eventually enter a bounded absorbing set. In particular, the associated omega-limit sets are compact and support a stationary statistical solution.
Low Mach number limit of some staggered schemes for compressible barotropic flows
Aix-Marseille Université, CNRS, École Centrale de Marseille（艾克斯-马赛大学、法国国家科学研究中心、马赛中央理工学院）；Institut de Radioprotection et de Sûreté Nucléaire (IRSN)（核辐射防护与核安全研究所）；Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan（里昂大学、法国国家科学研究中心、里昂第一大学、卡米尔·若尔当研究所）
The compressible barotropic Navier-Stokes type system in monodimensional case with Neumann boundary condition given on free boundary is considered. The local and the global existence with uniformly boundedness for small viscosity coefficient is proved.
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…
This paper establishes a blow-up criterion of strong solutions to the two-dimensional compressible magnetohydrodynamic (MHD) flows. The criterion depends on the density, but is independent of the velocity and the magnetic field. More precisely, once the strong solutions blow up, the $L^{\infty}$-norm for the density tends to infinity. In particular, the vacuum in the solutions is allowed.
It is shown that the one-dimensional or two-dimensional radially symmetric isothermal compressible Navier-Stokes system has no non-trivial global smooth solutions if the initial density is compactly supported. This result is a generalization of Xin's work \cite{Xin98} to the isothermal case.
We study the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums. We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group (see definition in the paper). The results hold regardless of either the size of the initial data or the far fields being vacuum or not. This improves the blowup results of Xin (1998) by removing the crucial assumptions that the initial density has compact support and the smooth solution h…
