Large-time Behavior Of Strong Solutions Research Articles

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.

Strong solutions to a nonlinear stochastic aggregation-diffusion equation

It is well-known that solutions to deterministic nonlocal aggregation-diffusion models may blow up in two or higher dimensions. Various mechanisms hence have been proposed to “regularize” the deterministic aggregation-diffusion equations in a manner that allows pattern formation without blow-up. However, stochastic effect has not been ever considered among other things. In this work, we consider a nonlocal aggregation-diffusion model with multiplicative noise and establish the local existence and uniqueness of strong solutions on [Formula: see text]. If the noise is non-autonomous and linear, we establish the global existence and large-time behavior of strong solutions with decay properties by combining the Moser-Alikakos iteration technique and some decay estimates of Girsanov type processes. If the noise is nonlinear and strong enough, we show that blow-up can be prevented. As such, our results assert that certain multiplicative noise can also regularize the aggregation-diffusion model.

THE ASYMPTOTIC BEHAVIOR OF STRONG SOLUTIONS TO THE CHEMOTAXIS MODEL IN THE CRITICAL FRAMEWORK

<abstract> The Keller-Segel model is an effective mathematical model (derived by Keller and Segel), which is used to describe the phenomenon of chemotaxis in biological sciences. The purpose of this paper is to investigate the asymptotic behavior of solutions in the $ L^p $ framework by developing the pure energy approach (independent of spectral analysis). Precisely, a new low-frequency regularity of initial data is posted, which enables us to establish the Lyapunov-type inequality in time for energy norms. As a result, the large-time behavior of strong solutions near constant equilibrium can be obtained. The proof crucially depends on non standard product estimates and interpolations. It's worth noting that the smallness requirement of low frequencies is no longer needed. </abstract>

Global regularity for a rapidly rotating constrained convection model of tall columnar structure with weak dissipation

We study a three-dimensional fluid model describing rapidly rotating convection that takes place in tall columnar structures. The purpose of this model is to investigate the cyclonic and anticyclonic coherent structures. Global existence, uniqueness, continuous dependence on initial data, and large-time behavior of strong solutions are shown provided the model is regularized by a weak dissipation term.

Global well-posedness and optimal large-time behavior of strong solutions to the non-isentropic particle-fluid flows

In this paper, we study the three-dimensional non-isentropic compressible fluid-particle flows. The system involves coupling between the Vlasov-Fokker-Planck equation and the non-isentropic compressible Navier-Stokes equations through momentum and energy exchanges. For the initial data near the given equilibrium we prove the global well-posedness of strong solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. For the periodic domain the same global well-posedness result still holds while the convergence rate is exponential. %The proof is based on the new macro-micro decomposition and energy estimates. New ideas and techniques are developed to establish the well-posedness and large-time behavior. For the global well-posedness our methods are based on the new macro-micro decomposition and fine energy estimates, while the proofs of the optimal large-time behavior rely on the Fourier analysis of the linearized Cauchy problem and the energy-spectrum method.

A remark on the time-decay estimates for the compressible magnetohydrodynamic system

ABSTRACT The present paper is dedicated to the optimal time-decay estimates of global strong solutions near constant equilibrium (away from vacuum) to the compressible magnetohydrodynamic (MHD) equations in the critical Besov spaces. In which we claim a new low-frequency assumption that plays a key role in the large-time behavior of solutions. Precisely, we exhibit that if the low frequencies of initial data belong to some Besov space with (), then the norm (the slightly stronger norm in fact) of strong solutions has the optimal decay ( if ) for , which improve the results of [Shi WX, Xu J. Large-time behavior of strong solutions to the compressible magnetohydrodynamic system in the critical framework. J Hyperbol Differ Equ. 2018;15:259–290]. The proof mainly depends on a sharp time-weighted energy estimates in light of low and high frequencies for the solutions. As a by-product, those optimal time-decay rates of - type are also captured in the critical framework.

Optimal Large-Time Behavior of the Two-Phase Fluid Model in the Whole Space

We investigate the large-time behavior of strong solutions to a two-phase fluid model in the whole space $\mathbb R^3$. This model was first derived by Choi [SIAM J. Math. Anal., 48 (2016), pp. 3090--3122] by taking the hydrodynamic limit from the Vlasov--Fokker--Planck/isentropic Navier--Stokes equations with strong local alignment forces. Under the assumption that the initial perturbation around an equilibrium state is sufficiently small, the global well-posedness issue has been established in [SIAM J. Math. Anal., 48 (2016), pp. 3090--3122]. However, as indicated by Choi, the large-time behavior of these solutions has remained an open problem. In this article, we resolve this problem by proving convergence to its associated equilibrium with the optimal rate which is the same as that of the heat equation. Particularly, the optimal convergence rates of the higher-order spatial derivatives of the solutions are also obtained. Moreover, for well-chosen initial data, we also show the lower bounds on the convergence rates. Our method is based on Hodge decomposition, low-frequency and high-frequency decomposition, delicate spectral analysis, and energy methods.

Global existence and large-time behavior of strong solutions to the Cauchy problem of the incompressible nematic liquid crystal flow

Global existence and large-time behavior of strong solutions to the Cauchy problem of the incompressible nematic liquid crystal flow

Nonlinearly exponential stability of compressible Navier-Stokes system with degenerate heat-conductivity

We study the large-time behavior of strong solutions to the one-dimensional, compressible Navier-Stokes system for a viscous and heat conducting ideal polytropic gas, when the viscosity is constant and the heat conductivity is proportional to a positive power of the temperature. Both the specific volume and the temperature are proved to be bounded from below and above independently of time. Moreover, it is shown that the global solution is nonlinearly exponentially stable as time tends to infinity. Note that the conditions imposed on the initial data are the same as those of the constant heat conductivity case (Kazhikhov-Shelukhin (1977) [18]; Kazhikhov (1981) [17]) and can be arbitrarily large. Therefore, our result can be regarded as a natural generalization of the Kazhikhov's ones for the constant heat conductivity case to the degenerate and nonlinear one.

Global dynamics of the thermomechanical Cucker–Smale ensemble immersed in incompressible viscous fluids

We study a global dynamics of the thermomechanical Cucker–Smale (TCS) particles immersed in incompressible viscous fluid. For this, we use a coupled system consisting of kinetic TCS equation for particles and incompressible Navier–Stokes for fluid via a drag force in three dimensions. We first show the global existence of weak solutions for large data, and then establish the global existence and uniqueness of strong solutions under a smallness assumption on the initial data. We also provide a priori estimate of large-time behavior of strong solutions which exhibits an exponential alignment between TCS particles and fluid asymptotically.

Large-time behavior of strong solutions to the compressible magnetohydrodynamic system in the critical framework

We study the compressible viscous magnetohydrodynamic (MHD) system and investigate the large-time behavior of strong solutions near constant equilibrium (away from vacuum). In the 80s, Umeda et al. considered the dissipative mechanisms for a rather general class of symmetric hyperbolic–parabolic systems, which is given by [Formula: see text] Here, [Formula: see text] denotes the characteristic root of linearized equations. From the point of view of dissipativity, Kawashima in his doctoral dissertation established the optimal time-decay estimates of [Formula: see text]-[Formula: see text]) type for solutions to the MHD system. Now, by using Fourier analysis techniques, we present more precise description for the large-time asymptotic behavior of solutions, not only in extra Lebesgue spaces but also in a full family of Besov norms with the negative regularity index. Precisely, we show that the [Formula: see text] norm (the slightly stronger [Formula: see text] norm in fact) of global solutions with the critical regularity, decays like [Formula: see text] as [Formula: see text]. Our decay results hold in case of large highly oscillating initial velocity and magnetic fields, which improve Kawashima’s classical efforts.

Large-time behavior of the strong solution to nonhomogeneous incompressible MHD system with general initial data

This paper investigates the large-time behavior of strong solutions to the nonhomogeneous incompressible magnetohydrodynamic equations on a bounded domain in $\mathbb{R}^{2}$ . Based on uniform estimates, we prove that the velocity, the magnetic field, and their derivatives converge to zero in $L^{2}$ norm as time goes to infinity without any additional assumption on the initial data and external force by a pure energy method.

Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation

In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic equation u t =−(u u xxx ) x , modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u 0(x)∈H 1(ℝ), both compactly supported or of finite second moment, we prove explicit and universal algebraic decay in the L 1-norm of the strong solution u(x,t) towards the unique (among source type solutions) strong source type solution of the equation with the same mass. The method we use is based on the study of the time decay of the entropy introduced in [13] for the porous medium equation, and uses analogies between the thin film equation and the porous medium equation.