Constant Equilibrium State Research Articles

Well-posedness and decay structure of a quantum hydrodynamics system with Bohm potential and linear viscosity

In this paper, a compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The purpose of this study is twofold. First, it is shown that the system is locally well-posed. For that purpose, the existence of classical solutions which are perturbation of constant states is established. Second, it is proved that in the particular case of subsonic equilibrium states, sufficiently small perturbations decay globally in time. In order to prove this stability property, the linearized system around the subsonic state is examined. Using an appropriately constructed compensating matrix symbol in the Fourier space, it is proved that solutions to the linear system decay globally in time, underlying a dissipative mechanism of regularity gain type. These linear decay estimates, together with the local existence result, imply the global existence and the decay of perturbations to constant subsonic equilibrium states as solutions to the full nonlinear system.

Finite-time blow-up in hyperbolic Keller–Segel system of consumption type with logarithmic sensitivity

This paper deals with finite-time blow-up of a hyperbolic Keller–Segel system of consumption type with the logarithmic sensitivity 0\\right)$?> ∂tρ=−χ∇⋅ρ∇logc,∂tc=−μcρχ,μ>0 in Rd(d⩾1) for nonvanishing initial data. This system is closely related to tumor angiogenesis, an important example of chemotaxis. Our singularity formation is not because c touches zero (which makes logc diverge) but due to the blowup of C1×C2 -norm of (ρ,c) . As a corollary, we also construct initial data near any constant equilibrium state which blows up in finite time for any d⩾1 .

Global non-relativistic quasi-neutral limit for a two-fluid Euler-Maxwell system

This paper concerns a two-fluid Euler-Maxwell system for plasmas with two small parameters. We study the non-relativistic quasi-neutral limit for smooth solutions of the system in whole space R3. When the initial data are smooth and sufficiently close to constant equilibrium states, we give a rigorous justification of the limit for all time. The limit system is governed by a compressible Euler system. The proof is based on uniform energy estimates and various dissipative estimates of smooth solutions with respect to the parameters and time. A key step in these estimates is to control the quasi-neutrality of the velocities by using a projection operator.

Global convergence of a two-fluid non-isentropic Euler–Poisson system in one space dimension

In this paper, we study the global in time quasi-neutral limit for a two-fluid non-isentropic Euler–Poisson system in one space dimension. We prove that the system converges to the non-isentropic Euler equations as the Debye length tends to zero. This problem is studied for smooth solutions near the constant equilibrium state. To prove this result, we establish uniform energy estimates and various dissipation estimates with respect to the Debye length and the time. These estimates allow to pass to the limit to obtain the limit system by compactness arguments. In addition, the global convergence rate is obtained by use of stream function technique.

Global existence and the time decay estimates of solutions to the compressible quantum Navier–Stokes–Maxwell system in R3$\mathbb {R}^3$

AbstractWe consider the Cauchy problem of the compressible quantum Navier–Stokes–Maxwell equations with the linear damping in the isentropic case under the small perturbation of the constant equilibrium state in three dimensions. Based on the refined energy method, we establish the classical solution globally in time in Sobolev space. By the combination of the energy estimates with the interpolation between the positive Sobolev norms and the negative Sobolev norms with , we also obtain the algebraic decay rates of the classical solution. What is more, the –L2 types of the time decay rates of the solution are obtained without small assumption on the initial data in L1 norm.

Global strong solutions to the one-dimensional compressible Navier-Stokes-Allen-Cahn system with phase field variable dependent viscosity

In this paper, we study the global existence and large-time behavior of strong solutions to the Cauchy problem of the one-dimensional isentropic compressible Navier-Stokes-Allen-Cahn system with phase field variable dependent viscosity, which models the motion of a mixture of two viscous compressible fluids. The case when the pressure p(ρ)=ργ, the viscosity ν(ρ,χ)=χα, the interface thickness δ(ρ)=ρβ and the relaxation time function a(ρ,χ,χy)=χλ is considered, where ρ>0 and χ are the density and the phase field variable, respectively, and γ,α,β,λ∈R are parameters. Under some conditions on the parameters γ,α,β,λ and the initial data, it is shown that the Cauchy problem of the one-dimensional isentropic compressible Navier-Stokes-Allen-Cahn system admits a unique global strong nonvacuum solution, which tends to the constant equilibrium state as time goes to infinity. Moreover, the exponential time decay rate of the phase field variable χ is also obtained. Here the initial perturbation for the phase field variable χ should be small, but the initial perturbations for the density and velocity of the fluid can be arbitrarily large.

Global solutions of 2D isentropic compressible Navier-Stokes equations with one slow variable

Motivated by [2,3], we prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which are slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to 2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of some constant equilibrium state nor of small energy.

Large-time behavior of solutions to three-dimensional bipolar Euler–Poisson equations with time-dependent damping

In this paper, we shall investigate the large-time behavior of solutions to the Cauchy problem for the three-dimensional bipolar isentropic Euler–Poisson equations with time-dependent damping effects for . Here, we consider a more general case that the two pressure functions are different and the doping profile is non-zero. Under the assumption that the initial data are close to the constant equilibrium states, we show that the smooth solutions to the Cauchy problem exist uniquely and globally. The algebraic time-decay rates of the solutions toward the constant equilibrium states are also obtained. The main ingredient of the proof is the time-weighted energy method with artfully chosen time-weighted functions.

On instability of a generic compressible two-fluid model in

This paper concerns the study of instability of a generic compressible two-fluid model in the whole space , where the capillary pressure is taken into account. For the case that the capillary pressure is a strictly decreasing function near the equilibrium, namely, , Evje et al (2016 Arch. Ration. Mech. Anal. 221 1285–316) established global stability of the constant equilibrium state for the three-dimensional Cauchy problem under some smallness assumptions. Recently, Wu et al (2022 arXiv:2204.10706) proved global stability of the constant equilibrium state for the case (corresponding to ). We investigate instability of the constant equilibrium state for the case that the capillary pressure is a strictly increasing function near the equilibrium, namely, . By employing the Hodge decomposition technique and making a detailed analysis of the Green’s function for the corresponding linearized system, we construct solutions of the linearized problem that grow exponentially in time in the Sobolev space Hk , thus leading to a global instability result for the linearized problem. Moreover, with the help of this global linear instability result together with a local existence theorem of classical solutions to the original nonlinear system, we can show instability of the nonlinear problem in the sense of Hadamard by carefully analysing the properties of the semigroup. Therefore, our results show that for the case , the constant equilibrium state of the two-fluid model is globally linearly unstable and locally nonlinearly unstable in the sense of Hadamard, which is different from the cases (Evje et al 2016 Arch. Ration. Mech. Anal. 221 1285–316) and (corresponding to ) (Wu et al 2022 arXiv:2204.10706) where the constant equilibrium state of the two-fluid model was proved to be globally nonlinearly stable.

Large-time behavior of global solutions to the Cauchy problem of one-dimensional viscous and heat-conducting ionized gas

This paper is concerned with the Cauchy problem of one-dimensional viscous and heat-conducting ionized gas when the viscosity is a positive constant or depends on the density. According to Saha's ionization equation, the equations of state for the basic thermodynamic variables of an ionized gas depend also on the degree of ionization, which leads to the loss of concavity of the physical entropy in some small bounded domain. Such a property of the ionized gas together with the unboundedness of the domain under our consideration makes it hard to derive the desired dissipative estimates on the first-order spatial derivative of both the bulk velocity and the absolute temperature, thus making the problem more challenging.In this paper, for a class of constant non-vacuum equilibrium states with positive temperature, we succeed in deducing the desired uniform-in-time bounds on the above mentioned dissipative estimates on both the bulk velocity and the absolute temperature. Based on such an estimate, we can then establish the global existence of solutions to the Cauchy problem of one-dimensional viscous and heat-conducting ionized gas with positive constant or density-dependent viscosity and show that such a class of constant non-vacuum equilibrium states is time asymptotically nonlinear stable.

Low Mach number limit of the global solution to the compressible Navier–Stokes system for large data in the critical Besov space

In this paper, we consider the compressible Navier–Stokes system around the constant equilibrium states and prove the existence of a unique global solution for arbitrarily large initial data in the scaling critical Besov space provided that the Mach number is sufficiently small, and the incompressible part of the initial velocity generates the global solution of the incompressible Navier–Stokes equation. Moreover, we consider the low Mach number limit and show that the compressible solution converges to the solution of the incompressible Navier–Stokes equation in some space time norms.

Global Well-Posedness and Large-Time Behavior for the Equilibrium Diffusion Model in Radiation Hydrodynamics

In this paper, we investigate the global well-posedness of the equilibrium diffusion model in radiation hydrodynamics. The model consists of the compressible Navier-Stokes equations coupled with radiation effect terms described by the fourth power of temperature. The global existence of classical solutions to the Cauchy problem in the whole space is established when initial data is a small smooth perturbation of a constant equilibrium state: moreover, an algebraic rate of convergence of solutions toward equilibrium is obtained under additional conditions on initial data. The proof is based on the refined energy method and Fourier’s analysis.

Large time behavior of the Euler-Poisson system coupled to a magnetic field

<abstract><p>In this paper, the large time behavior of globally smooth solutions of the Cauchy problem for the three dimensional Euler-Poisson system of compressible fluids coupled to a magnetic field is studied. We prove that the smooth solutions (near a given constant equilibrium state) of the problem converge asymptotically to a stationary solution exponentially fast as $ t $ goes to $ \infty $.</p></abstract>

Global existence of smooth solutions to a full Euler–Poisson system in one space dimension

When a strictly dissipative term arises in the energy equation, the Cauchy problem for Euler–Poisson systems admits global smooth solutions for small initial data. This was proved in previous studies. In this paper, we study a stability problem for a full Euler–Poisson system without dissipation in the energy equation. We prove the global existence of smooth solutions near constant equilibrium states in one space dimension. The stability results are obtained for both one-fluid and two-fluid Euler–Poisson systems. In our proof of these results, we show an L2 energy equality in Euler coordinates. Then, we establish energy estimates of derivatives of the solution in Lagrangian coordinates by a characteristic technique. These estimates together with the equivalence of smooth solutions between the two coordinates yield the global existence of the solution.

Critical regularity issues for the compressible Navier–Stokes system in bounded domains

We are concerned with the barotropic compressible Navier–Stokes system in a bounded domain of mathbb {R}^d (with dge 2). In a critical regularity setting, we establish local well-posedness for large data with no vacuum and global well-posedness for small perturbations of a stable constant equilibrium state. Our results rely on new maximal regularity estimates—of independent interest—for the semigroup of the Lamé operator, and of the linearized compressible Navier–Stokes equations.

Global quasi-neutral limit for a two-fluid Euler-Poisson system in one space dimension

The quasi-neutral limit of one-fluid Euler-Poisson systems leads to incompressible Euler equations. It was widely studied in previous works. In this paper, we deal with the quasi-neutral limit in a two-fluid Euler-Poisson system. This limit presents a different feature since it leads to compressible Euler equations. We justify this limit for global smooth solutions near constant equilibrium states in one space dimension. Specifically, we prove a global existence of smooth solutions by establishing uniform energy estimates with respect to the Debye length and the time. This allows to pass to the limit in the system for all time. Moreover, we establish global error estimates between the solution of the two-fluid Euler-Poisson system and that of the compressible Euler equations. The proof is based on classical uniform energy estimates together with various dissipation estimates. In order to control the quasi-neutrality of the velocities of two-fluids, similar conditions on the initial data are needed.

Dissipative structure of one-dimensional isothermal compressible fluids of Korteweg type

This paper studies the dissipative structure of the system of equations that describes the motion of a compressible, isothermal, viscous-capillar fluid of Korteweg type in one space dimension. It is shown that the system satisfies the genuine coupling condition of Humpherys [23], which is, in turn, an extension to higher order systems of the classical condition by Kawashima and Shizuta [30,42] for second order systems. It is proved that genuine coupling implies the decay of solutions to the linearized system around a constant equilibrium state. For that purpose, the symmetrizability of the Fourier symbol is used in order to construct an appropriate compensating matrix. These linear decay estimates imply the global decay of perturbations to constant equilibrium states as solutions to the full nonlinear system, via a standard continuation argument.

Global well-posedness and time-decay of solutions for the compressible Hall-magnetohydrodynamic system in the critical Besov framework

We consider the global existence of solution for the initial value problem for the compressible Hall-magnetohydrodynamic system in the whole space R3. The system consists of a hyperbolic-parabolic system of partial differential equations of the conservation laws type with non-symmetric diffusion. We show the existence of solution as a perturbation from a constant equilibrium state (ρ¯,0,B¯), where ρ¯>0 is a constant density, 0∈R3 is the zero velocity and B¯∈R3 is a constant magnetic field. The time-decay of the solution in the Besov spaces is also established. Our results show the pointwise estimate of the solution in the Fourier space for the linearized Hall-MHD system that related to the result obtained by Umeda–Kawashima–Shizuta [33] for a general class of linear symmetric hyperbolic-parabolic systems with symmetric diffusion. We utilize a systematic use of the product estimates in the Chemin–Lerner spaces and apply the energy method due to Matsumura–Nishida [27].

On the initial-boundary value problem for the three-dimensional compressible viscoelastic fluids with the electrostatic effect

We investigate the initial-boundary value problem of the three-dimensional compressible viscoelastic fluids with the electrostatic effect, in which the Dirichlet-Neumann mixed boundary condition for the electrostatic potential is imposed. We prove that there exists a unique global-in-time small strong solution in H2 Sobolev space. Moreover, we show that such a solution converges to the constant equilibrium state with an exponential decay rate as time tends to infinity.

Global well-posedness and large-time behavior of a hyperbolic-parabolic model in an exterior domain

<p style='text-indent:20px;'>We consider a hyperbolic-parabolic system derived from a limiting case of chemotaxis model in an exterior domain. By energy method we prove global existence for the initial-boundary value problem in an exterior domain when the initial data is a small perturbation around constant equilibrium states, and the solutions converge to the states at an algebraic decay rate as time goes to infinity.</p>