Constant Equilibrium State Research Articles

<i>L</i><sup><i>P</i></sup> decay for general hyperbolic-parabolic systems of balance laws

We study time asymptotic decay of solutions for a general system of hyperbolic-parabolic balance laws in multi space dimensions. The system has physical viscosity matrices and a lower order term for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper we obtain optimal $L^p$ decay rates for $p≥2$. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively.

Convergence of a non-isentropic Euler–Poisson system for all time

We consider periodic smooth solutions for a non-isentropic Euler–Poisson system with small parameters, in which the momentum and energy equations are partially dissipative. When initial data are close to constant equilibrium states, we prove the global-in-time convergence of the system as parameters go to zero. The limit systems are incompressible Euler equations with dissipation, the drift-diffusion system and the energy-transport system. The proof of the results is based on compactness arguments and uniform energy estimates with respect to the time and the parameters.

Global existence theory for general hyperbolic-parabolic balance laws with application

We study a general system of hyperbolic-parabolic balance laws in [Formula: see text] space dimensions ([Formula: see text]). The system has rank deficient viscosity matrices and a lower order term whose Jacobian matrix is rank deficient as well. We consider the Cauchy problem when initial data are small perturbations of a constant equilibrium state. Under a set of reasonable assumptions including Kawashima–Shizuta condition, we establish the existence of solution global in time via energy method. The proposed assumptions are sufficiently general for applications to physical models such as electro-magneto flows and physical gas flows. In particular, we study the gas flow with an internal non-equilibrium mode besides the translational non-equilibrium. The general result in this paper recovers the existing results in literature on hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively, as two special cases.

Galactic cold cores

Filaments are key for star formation models. As part of the study carried out by the Herschel GCC Programme, here we study the filament properties presented in GCC.VII in context with theoretical models of filament formation and evolution. A conservative sample of filaments at a distance D<500pc was extracted with the Getfilaments algorithm. Their physical structure was quantified according to two main components: the central (Gaussian) region (core component), and the power-law like region dominating the filament column density profile at larger radii (wing component). The properties and behaviour of these components relative to the total linear mass density of the filament and its environmental column density were compared with theoretical models describing the evolution of filaments under gravity-dominated conditions. The feasibility of a transition to supercritical state by accretion is dependent on the combined effect of filament intrinsic properties and environmental conditions. Reasonably self-gravitating (high Mline-core) filaments in dense environments (av\sim3mag) can become supercritical in timescales of t\sim1Myr by accreting mass at constant or decreasing width. The trend of increasing Mline-tot (Mline-core and Mline-wing), and ridge Av with background also indicates that the precursors of star-forming filaments evolve coevally with their environment. The simultaneous increase of environment and filament Av explains the association between dense environments and high Mline-core values, and argues against filaments remaining in constant single-pressure equilibrium states. The simultaneous growth of filament and background in locations with efficient mass assembly, predicted in numerical models of collapsing clouds, presents a suitable scenario for the fulfillment of the combined filament mass-environment criterium that is in quantitative agreement with Herschel observations.

Global solutions to the equation of thermoelasticity with fading memory

We consider the initial-history value problem for the one-dimensional equation of thermoelasticity with fading memory. It is proved that if the data are smooth and small, then a unique smooth solution exists globally in time and converges to the constant equilibrium state as time goes to infinity. Our proof is based on a technical energy method which makes use of the strict convexity of the entropy function and the properties of strongly positive definite kernels.

Global Existence and the Optimal Decay Rates for the Three Dimensional Compressible Nematic Liquid Crystal Flow

The present paper is dedicated to the study of the Cauchy problems for the three-dimensional compressible nematic liquid crystal flow. We obtain the global existence and the optimal decay rates of smooth solutions to the system under the condition that the initial data in lower regular spaces are close to the constant equilibrium state. Our main method is based on the spectral analysis and the smooth effect of dissipative operator.

Global quasi-neutral limit of Euler–Maxwell systems with velocity dissipation

We consider smooth solutions to the Cauchy problem for an isentropic Euler–Maxwell system with velocity dissipation and small physical parameters. For initial data uniformly close to constant equilibrium states, we prove the global-in-time convergence of the system as the parameters go to zero. The limit systems are the e-MHD system and the incompressible Euler equations, both with velocity dissipation. The proof of the results relies on a single uniform energy estimate with respect to the time and the parameters, together with compactness arguments. For this purpose, the classical energy estimates for the symmetrizable hyperbolic system are not sufficient. We construct a Lyapunov type energy by controlling the divergence and the curl of the velocity, the electric and magnetic fields.

Uniform global existence and convergence of Euler-Maxwell systems with small parameters

The Euler-Maxwell system with small parameters arises in the modeling of magnetized plasmas and semiconductors. For initial data close to constant equilibrium states, we prove uniform energy estimates with respect to the parameters, which imply the global existence of smooth solutions. Under reasonable assumptions on the convergence of initial conditions, this allows to show the global-in-time convergence of the Euler-Maxwell system as each of the parameters goes to zero.

Optimal time decay rate for compressible viscoelastic equations in critical spaces

In this paper, we are concerned with the optimal time convergence rate of the global strong solution to some constant equilibrium states for the compressible viscoelastic fluids in the whole space. Green’s matrix method and energy estimate method are used to obtain the optimal time decay rate under the critical Besov space framework. Our result implies the optimal -time decay rate and only need the initial datum to be small in some critical Besov space which have very low regularity compared with the classical Sobolev space.

Long-time behavior of solution for the compressible nematic liquid crystal flows in [formula omitted

In this paper, we investigate the global existence and long-time behavior of classical solution for the compressible nematic liquid crystal flows in three-dimensional whole space. First of all, the global existence of classical solution is established under the condition that the initial data are close to the constant equilibrium state in HN(R3) (N≥3)-framework. Then, one establishes algebraic time decay for the classical solution by weighted energy method. Finally, the algebraic decay rate of classical solution in Lp(R3)-norm with 2≤p≤∞ and optimal decay rate of their spatial derivative in L2(R3)-norm are obtained if the initial perturbation belong to L1(R3) additionally.

Global existence and large time behavior of the asymmetric fluids

In this paper, we consider the asymptotic stability of the steady state with the constant equilibrium state. Under the assumptions that the \({H^3}\) norm of the initial data is small, but its higher-order derivatives could be large, we prove the global existence to the Cauchy problem for the asymmetric fluids in \({\mathbb{R}^3}\). Moreover, we obtain the time decay rates of the solutions and their higher-order spatial derivatives by introducing the negative Sobolev and Besov spaces.

Global existence and decay rate of the Boussinesq–Burgers system with large initial data

In this paper, we study the global existence and the long-time asymptotic behavior of classical solutions to the Cauchy problem for the Boussinesq–Burgers system with large initial data. More precisely, we first show that the classical solutions exist globally in time with large initial data by using the Lp (p>2) estimates other than the conventional L2 estimate. Then we prove that the global classical solutions converge to constant equilibrium states with an algebraic decay rate as time approaches infinity.

Uniform global existence and parabolic limit for partially dissipative hyperbolic systems

This work concerns smooth solutions to the Cauchy problem for first-order partially dissipative hyperbolic systems with a small parameter. The systems are written in non-conservative form in several space variables. We introduce algebraic conditions on the structure of the systems. Under these conditions together with a partial dissipation condition and the Shizuta–Kawashima stability condition, we prove three main results around constant equilibrium states. These results are uniform global existence with respect to the parameter, global-in-time convergence of the systems to second-order nonlinear parabolic systems in a slow time variable, and global existence when the parameter is fixed. We also give examples of physical models to which the above results can be applied.

Pointwise time asymptotic behavior of solutions to a general class of hyperbolic balance laws

In this paper we are interested in a general class of hyperbolic balance laws. Within the frame work of existence of a convex entropy function that symmetrizes the system in certain sense and the Kawashima–Shizuta condition, we study the large time behavior of the solution to the Cauchy problem in one space dimension. For a solution around a constant equilibrium state, we predetermine the asymptotic solution as a superposition of the constant state and diffusion waves along the equilibrium characteristic directions. We estimate the remainder in the pointwise sense both in space and in time. The decay rates in various directions are optimal, which further give the optimal Lp rates with 1≤p≤∞. Applications to several examples including the Kerr–Debye model are given.

A Generalized Poisson--Nernst--Planck--Navier--Stokes Model on the Fluid with the Crowded Charged Particles: Derivation and Its Well-Posedness

We derive a hydrodynamic model of the compressible conductive fluid by using an energetic variational approach, which could be called a generalized Poisson--Nernst--Planck--Navier--Stokes system. This system characterizes the micro-macro interactions of the charged fluid and the mutual friction between the crowded charged particles. In particular, it reveals the cross-diffusion phenomenon which does not happen in the fluid with the dilute charged particles. The cross-diffusion is tricky; however, we develop a general method to show that the system is globally asymptotically stable under small perturbations around a constant equilibrium state. Under some conditions, we also obtain the optimal decay rates of the solution and its derivatives of any order. Our method will apply equally well to a class of cross-diffusion systems if their linearized diffusion matrices are diagonally dominant.

Global existence and optimal decay rates of solutions for compressible Hall-MHD equations

In this paper, we are concerned with global existence and optimal decay rates ofsolutions for the compressible Hall-MHD equations in dimension three.First, we prove the global existence ofstrong solutions by the standard energy method under the conditionthat the initial data are close to the constant equilibrium state in $H^2$-framework.Second, optimal decay rates of strong solutions in $L^2$-norm are obtainedif the initial data belong to $L^1$ additionally.Finally, we apply Fourier splitting method bySchonbek [Arch. Rational Mech. Anal. 88 (1985)] to establish optimal decay rates forhigher order spatial derivatives of classical solutions in $H^3$-framework,which improves the work of Fan et al.[Nonlinear Anal. Real World Appl. 22 (2015)].

Analysis of non-isentropic compressible Euler equations with relaxation

This paper is contributed to the study of the one-dimensional non-isentropic compressible Euler equations with relaxation. It is shown that classical solutions do not exist globally-in-time under general conditions on initial data. Indeed, finite-time blowup occurs in a quantity related to the first moment. On the other hand, when the initial datum is sufficiently close to a constant equilibrium state, it is shown that the equations possess a unique global-in-time classical solution, and the solution converges to the equilibrium state in the long-time run. When the domain is finite, the convergence rate is shown to be exponential, due to boundary effects.

Darcy's law and diffusion for a two-fluid Euler–Maxwell system with dissipation

This paper is concerned with the large-time behavior of solutions to the Cauchy problem on the two-fluid Euler–Maxwell system with dissipation when initial data are around a constant equilibrium state. The main goal is the rigorous justification of diffusion phenomena in fluid plasma at the linear level. Precisely, motivated by the classical Darcy's law for the nonconductive fluid, we first give a heuristic derivation of the asymptotic equations of the Euler–Maxwell system in large time. It turns out that both the density and the magnetic field tend time-asymptotically to the diffusion equations with diffusive coefficients explicitly determined by given physical parameters. Then, in terms of the Fourier energy method, we analyze the linear dissipative structure of the system, which implies the almost exponential time-decay property of solutions over the high-frequency domain. The key part of the paper is the spectral analysis of the linearized system, exactly capturing the diffusive feature of solutions over the low-frequency domain. Finally, under some conditions on initial data, we show the convergence of the densities and the magnetic field to the corresponding linear diffusion waves with the rate [Formula: see text] in L2-norm and also the convergence of the velocities and the electric field to the corresponding asymptotic profiles given in the sense of the generalized Darcy's law with the faster rate [Formula: see text] in L2-norm. Thus, this work can be also regarded as the mathematical proof of the Darcy's law in the context of collisional fluid plasma.

Quantitative decay of a one-dimensional hybrid chemotaxis model with large data

We investigate the quantitative behaviour of one-dimensional classical solutions for a hyperbolic-parabolic system describing repulsive chemotaxis. It is shown that classical solutions to the Cauchy problem always exist globally in time for large initial perturbations around constant equilibrium states. We prove rigorously the chemo-repulsion collapse scenario and show that all solutions converge to the attractive ground states as time approaches infinity. Moreover explicit decay rates of the perturbations are computed under some mild conditions on the initial data. The proof is established via a novel Lp-based energy method. We also obtain a frequency-dependent stretched-exponential decay rate by using a new Fourier method which can be of independent interest.

Thermal Non-Equilibrium Flows in Three Space Dimensions

We study the equations describing the motion of a thermal non-equilibrium gas in three space dimensions. It is a hyperbolic system of six equations with a relaxation term. The dissipation mechanism induced by the relaxation is weak in the sense that the Shizuta-Kawashima criterion is violated. This implies that a perturbation of a constant equilibrium state consists of two parts: one decays in time while the other stays. In fact, the entropy wave grows weakly along the particle path as the process is irreversible. We study thermal properties related to the well-posedness of the nonlinear system. We also obtain a detailed pointwise estimate on the Green’s function for the Cauchy problem when the system is linearized around an equilibrium constant state. The Green’s function provides a complete picture of the wave pattern, with an exact and explicit leading term. Comparing with existing results for one dimensional flows, our results reveal a new feature of three dimensional flows: not only does the entropy wave not decay, but the velocity also contains a non-decaying part, strongly coupled with its decaying one. The new feature is supported by the second order approximation via the Chapman-Enskog expansions, which are the Navier-Stokes equations with vanished shear viscosity and heat conductivity.