Constant Equilibrium State Research Articles

Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system

<p style='text-indent:20px;'>Recently, the authors proved [<xref ref-type="bibr" rid="b2">2</xref>] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [<xref ref-type="bibr" rid="b2">2</xref>]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.</p>

Convergence rates in zero-relaxation limits for Euler-Maxwell and Euler-Poisson systems

It was proved that Euler-Maxwell systems converge globally-in-time to drift-diffusion systems in a slow time scaling, as the relaxation time goes to zero. The convergence was established to the Cauchy problem with smooth periodic initial data sufficiently close to constant equilibrium states. In this paper, we establish error estimates between smooth periodic solutions of Euler-Maxwell systems and those of drift-diffusion systems. Similar error estimates are also obtained for Euler-Poisson systems in place of Euler-Maxwell systems. The proof of these results uses stream function techniques together with energy estimates.

Optimal time decay of the compressible Navier–Stokes equations for a reacting mixture

In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the compressible Navier–Stokes equations for ideal reacting gases. The asymptotic stability of the constant equilibrium state with strictly positive constant density, temperature and the vanishing velocity, mass fraction of the reactant is established under suitable small initial perturbation in . Precisely, we show the convergence of the density, velocity and temperature towards the corresponding equilibrium state with the optimal rate in L 2-norm as well as the convergence of the mass fraction to the equilibrium state with the optimal rate in L 2-norm. Furthermore, the optimal decay rates for the spatial-derivatives of the solution are also obtained. The proof is based on the time-weighted energy estimate and continuation argument.

Decay of strong solution for the compressible Navier–Stokes equations with large initial data

In this paper, we investigate the convergence of the global large solution to its associated constant equilibrium state with an explicit decay rate for the compressible Navier–Stokes equations in three-dimensional whole space. Suppose the initial data belongs to some negative Sobolev space instead of Lebesgue space, we not only prove the negative Sobolev norms of the solution being preserved along time evolution, but also obtain the convergence of the global large solution to its associated constant equilibrium state with algebra decay rate. Besides, we shall show that the decay rate of the first order spatial derivative of large solution of the full compressible Navier–Stokes equations converging to zero in L2−norm is (1+t)−54, which coincides with the heat equation. This extends the previous decay rate (1+t)−34 obtained in He et al. (2020).

Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations

Abstract We study the initial-boundary value problems of the three-dimensional compressible elastic Navier-Stokes-Poisson equations under the Dirichlet or Neumann boundary condition for the electrostatic potential. The unique global solution near a constant equilibrium state in H 2 space is obtained. Moreover, we prove that the solution decays to the equilibrium state at an exponential rate as time tends to infinity. This is the first result for the three-dimensional elastic Navier-Stokes-Poisson equations under various boundary conditions for the electrostatic potential.

Decay estimates of solutions to the compressible micropolar fluids system in R3

This work is concerned with the compressible micropolar fluids system in three–dimensional space. We consider the asymptotic behavior of the solution to the Cauchy problem near the constant equilibrium state provided that the initial perturbation is sufficiently small. Under some assumptions of the initial data, we show that the solution of the Cauchy problem converges to its constant equilibrium state at the exact same L 2 –decay rates as the linearized equations, which shows the convergence rates are optimal. The proof is based on the spectral analysis of the semigroup generated by the linearized equations and the nonlinear energy estimates.

Compressible Navier-Stokes approximation for the Boltzmann equation in bounded domains

It is well known that the full compressible Navier-Stokes equations can be deduced via the Chapman-Enskog expansion from the Boltzmann equation as the first-order correction to the Euler equations with viscosity and heat-conductivity coefficients of order of the Knudsen number $\epsilon>0$. In the paper, we carry out the rigorous mathematical analysis of the compressible Navier-Stokes approximation for the Boltzmann equation regarding the initial-boundary value problems in general bounded domains. The main goal is to measure the uniform-in-time deviation of the Boltzmann solution with diffusive reflection boundary condition from a local Maxwellian with its fluid quantities given by the solutions to the corresponding compressible Navier-Stokes equations with consistent non-slip boundary conditions whenever $\epsilon>0$ is small enough. Specifically, it is shown that for well chosen initial data around constant equilibrium states, the deviation weighted by a velocity function is $O(\epsilon^{1/2})$ in $L^\infty_{x,v}$ and $O(\epsilon^{3/2})$ in $L^2_{x,v}$ globally in time. The proof is based on the uniform estimates for the remainder in different functional spaces without any spatial regularity. One key step is to obtain the global-in-time existence as well as uniform-in-$\epsilon$ estimates for regular solutions to the full compressible Navier-Stokes equations in bounded domains when the parameter $\epsilon>0$ is involved in the analysis.

Global Existence and Long-Time Behavior of Solutions to the Full Compressible Euler Equations with Damping and Heat Conduction in ℝ 3

We study the Cauchy problem of the three-dimensional full compressible Euler equations with damping and heat conduction. We prove the existence and uniqueness of the global small H N N ≥ 3 solution; in particular, we only require that the H 4 norms of the initial data be small when N ≥ 5 . Moreover, we use a pure energy method to show that the global solution converges to the constant equilibrium state with an optimal algebraic decay rate as time goes to infinity.

Decay-in-time of the highest-order derivatives of solutions for the compressible isentropic MHD equations

In this paper, we consider the large time behavior of the Cauchy problem for the three-dimensional isentropic compressible magnetohydrodynamic (MHD) equations. The global existence of smooth solutions for the 3D compressible MHD equations has been proved by Chen–Tan [6], under the condition that the initial data are close to the constant equilibrium state in the Sobolev space H3. However, to our best knowledge, the decay estimate of the highest-order spatial derivatives of the solution to the compressible MHD equations has not been solved. The main goal in this paper is to give a positive answer to this problem. Exactly, under the assumption that the initial perturbation is small in Hl(R3)∩B˙2,∞−s(R3) with l⩾3, s∈[0,52], combining the spectral analysis on the semigroup generated by the linear system at the constant state and the energy method to the compressible MHD equations, then we get the optimal convergence rates of any order spatial derivatives (including the highest-order derivatives) of the solution. This result concern with the optimal time decay rates of the solutions, which extends the work obtained by Chen [5] for the compressible Navier–Stokes equations in R3 to the 3D compressible MHD equations. Moreover, we have expanded the range of s from (0,3/2] to [0,5/2], compared with the previous results on optimal decay rates of global strong solutions for the 3D isentropic compressible MHD system.

The global convergence of non-isentropic Euler–Maxwell equations via Infinity-Ion-Mass limit

This paper is concerned with the periodic problem to the two-fluid non-isentropic Euler–Maxwell (N-E-M) equations. The equations arises in the modeling of magnetic plasma, in which appear two physical parameters, the mass of an electron $$m_\mathrm{e}$$ and the mass of an ion $$m_{\mathrm{i}}$$ . With the help of methods of asymptotic expansions, we prove the local-in-time convergence of smooth solutions to this problem by setting $$m_\mathrm{e} = 1$$ and letting $$m_{\mathrm{i}} \rightarrow +\infty $$ . Moreover, when the initial data are near constant equilibrium states, by means of uniform energy estimates and compactness arguments, we rigorously prove the infinity-ion-mass convergence of the system for all time. The limit system is the one-fluid N-E-M system.

Large time behavior and diffusion limit for a system of balance laws from chemotaxis in multi-dimensions

We consider the Cauchy problem for a system of balance laws derived from a chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove the global well-posedness of classical solutions to the Cauchy problem when only the energy of the first order spatial derivatives of the initial data is sufficiently small, and the solutions are shown to converge to the prescribed constant equilibrium states as time goes to infinity. Then we prove that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model when the chemical diffusion coefficient tends to zero.

Convergence rate from systems of balance laws to isotropic parabolic systems, a periodic case

It is proved that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish the global-in-time error estimates between the smooth solutions to the partially dissipative hyperbolic systems and those to the isotropic parabolic limiting systems in a three dimensional torus, rather than in the one dimensional whole space (Appl. Anal. 100(5) (2021) 1079–1095). This avoids the condition raised for the strong connection between the flux and the source term and make the result obtained more generalized. In the proof, we provide a similar stream function technique which is valid for the three dimensional periodic case. Similar method is provided for the one-dimensional periodic case. As applications of the results, we give several examples arising from physical models at the end of the paper.

Global weak solutions and asymptotics of a singular PDE-ODE Chemotaxis system with discontinuous data

This paper is concerned with the well-posedness and large-time behavior of a two-dimensional PDE-ODE hybrid chemotaxis system describing the initiation of tumor angiogenesis. We first transform the system via a Cole-Hopf type transformation into a parabolic-hyperbolic system and then show that the solution of the transformed system converges to a constant equilibrium state as time tends to infinity. Finally we reverse the Cole-Hopf transformation and obtain the relevant results for the pre-transformed PDE-ODE hybrid system. In contrast to the existing related results, where continuous initial data is imposed, we are able to prove the asymptotic stability for discontinuous initial data with large oscillations. The key ingredient in our proof is the use of the so-called “effective viscous flux”, which enables us to obtain the desired energy estimates and regularity. The technique of the “effective viscous flux” turns out to be a very useful tool to study chemotaxis systems with initial data having low regularity and was rarely (if not) used in the literature for chemotaxis models.

Lower bound of decay rate for higher order derivatives of solution to the compressible quantum magnetohydrodynamic model

The lower bounds of decay rates for global solution to the compressible viscous quantum magnetohydrodynamic model in three‐dimensional whole space under the H5 × H4 × H4 framework are investigated in this paper. We first show that the lower bound of decay rate for the solution converging to constant equilibrium state (1, 0, 0) in L2‐norm is when the initial data satisfy some low‐frequency assumption. Moreover, we prove that the lower bound of decay rate of k(k ∈ [1, 3]) order spatial derivative for the solution converging to constant equilibrium state (1, 0, 0) in L2‐norm is . Then, we show that the lower bound of decay rate for the time derivatives of density and velocity is , but the lower bound of decay rate for the time derivative of magnetic field converging to zero in L2‐norm is .

The rigorous derivation of unipolar Euler–Maxwell system for electrons from bipolar Euler–Maxwell system by infinity‐ion‐mass limit

In the paper, we consider the local‐in‐time and the global‐in‐time convergence of infinity‐ion‐mass limit for bipolar Euler–Maxwell systems by setting the mass of an electron and letting the mass of an ion mi → +∞. We use the method of asymptotic expansions to handle the local‐in‐time convergence problem and find that the limiting process from bipolar models to unipolar models is actually decoupling but not the vanishing of equations for the corresponding the other particle. Moreover, when the initial data are sufficiently close to the constant equilibrium state, we also establish the corresponding global‐in‐time convergence.

Relaxed Euler systems and convergence to Navier-Stokes equations

We consider the approximation of Navier-Stokes equations for a Newtonian fluid by Euler type systems with relaxation both in compressible and incompressible cases. This requires to decompose the second-order derivative terms of the velocity into first-order ones. Usual decompositions lead to approximate systems with tensor variables. We construct approximate systems with vector variables by using Hurwitz-Radon matrices. These systems are written in the form of balance laws and admit strictly convex entropies, so that they are symmetrizable hyperbolic. For smooth solutions, we prove the convergence of the approximate systems to the Navier-Stokes equations in uniform time intervals. Global-in-time convergence is also shown for the initial data near constant equilibrium states of the systems. These convergence results are established not only for the approximate systems with vector variables but also for those with tensor variables.

Spatiotemporal patterns in a diffusive predator-prey model with protection zone and predator harvesting

Abstract In this paper, a diffusive predator-prey model subject to the zero flux boundary conditions is considered, in which the prey population exhibits social behavior and the harvesting functional of the predator population is assumed to be considered in a quadratic form. The existence of a positive solution and its bounders is investigated. The global stability of the semi trivial constant equilibrium state is established. Concerning the non trivial equilibrium state, the local stability, Hopf bifurcation, diffusion driven instability, Turing-Hopf bifurcation are investigated. The direction and the stability of Hopf bifurcation relying on the system parameters is derived. Some numerical simulations are used to extend the analytical results and show the occurrence of the homogeneous and non homogeneous periodic solutions. Further the effect of the rivalry rate on the dynamical behavior of the studied species.

Lower bound of decay rate for higher-order derivatives of solution to the compressible fluid models of Korteweg type

This paper concerns the lower bound decay rate of global solution for compressible Navier–Stokes–Korteweg system in three-dimensional whole space under the H^{4}times H^{3} framework. At first, the lower bound of decay rate for the global solution converging to constant equilibrium state (1, 0) in L^2-norm is (1+t)^{-frac{3}{4}} if the initial data satisfy some low-frequency assumption additionally. Furthermore, we also show that the lower bound of the k(kin [1, 3])th-order spatial derivatives of solution converging to zero in L^2-norm is (1+t)^{-frac{3+2k}{4}}. Finally, it is proved that the lower bound of decay rate for the time derivatives of density and velocity converging to zero in L^2-norm is (1+t)^{-frac{5}{4}}.

The optimal decay rate of strong solution for the compressible Navier–Stokes equations with large initial data

In a recent paper (He et al., 2019), it is shown that the upper decay rate of global solution of compressible Navier–Stokes(CNS) equations converging to constant equilibrium state (1,0) in H1−norm is (1+t)−34(2p−1) when the initial data is large and belongs to H2(R3)∩Lp(R3)(p∈[1,2)). Thus, the first result in this paper is devoted to showing that the upper decay rate of the first order spatial derivative converging to zero in H1−norm is (1+t)−32(1p−12)−12. For the case of p=1, the lower bound of decay rate for the global solution of CNS equations converging to constant equilibrium state (1,0) in L2−norm is (1+t)−34 if the initial data satisfies some low frequency assumption additionally. In other words, the optimal decay rate for the global solution of CNS equations converging to constant equilibrium state in L2−norm is (1+t)−34 although the associated initial data is large.

Decay estimates of solutions to the bipolar compressible Euler–Poisson system in $$\pmb {\mathbb {R}^3}$$

This work is concerned with the bipolar compressible Euler–Poisson equations with damping in three-dimensional space. We consider the optimal decay rates of the solution to the Cauchy problem, provided that the initial perturbation at the constant equilibrium state is sufficiently small. Under some assumptions of the initial data, we show that the solution to the Cauchy problem converges to its constant equilibrium state at the exact same $$L^2$$ decay rates as the linearized equations, which shows the convergence rates are sharp. The proof is based on the spectral analysis of the semigroup generated by the linearized equations and the nonlinear energy estimates.