Time Decay Rates Of Solutions Research Articles

Large Time Behavior and Stability for Two-Dimensional Magneto-Micropolar Equations with Partial Dissipation

This paper is devoted to the stability and decay estimates of solutions to the two-dimensional magneto-micropolar fluid equations with partial dissipation. Firstly, focus on the 2D magneto-micropolar equation with only velocity dissipation and partial magnetic diffusion, we obtain the global existence of solutions with small initial in H^s({mathbb {R}}^2)(s>1), and by fully exploiting the special structure of the system and using the Fourier splitting methods, we establish the large time decay rates of solutions. Secondly, when the magnetic field has partial dissipation, we show the global existence of solutions with small initial data in dot{B}^0_{2,1}({mathbb {R}}^2). In addition, we explore the decay rates of these global solutions are correspondingly established in dot{B}^m_{2,1}({mathbb {R}}^2) with 0 le m le s, when the initial data belongs to the negative Sobolev space dot{H}^{-l}({mathbb {R}}^2) (for each 0 le l <1).

Long time decay analysis of complex-valued fractional abstract evolution equations with delay

The asymptotic stability and long time decay rates of solutions to linear Caputo time-fractional ordinary differential equations are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of time-fractional ordinary differential equations decay only polynomially, leading to the so-called Mittag-Leffler stability, which were already extended to linear fractional delay differential equations (FDDEs) with real constants by Čermák et al. (2017) [17]. This paper is devoted to the asymptotical stability and long time decay rates of the solutions for a class of fractional delay abstract evolution equations with complex-valued coefficients. By applying the spectral decomposition method, we reformulate the fractional delay abstract evolution equations to an infinite-dimensional FDDEs. Through the root locus technique, we develop a necessary and sufficient stability condition in a coefficient criterion for FDDEs. At the same time, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long time decay rates of the solutions for both linear FDDEs and fractional delay abstract evolution equations with complex-valued coefficients are O(t−α). Finally, several application models including both time and time-space fractional Schrödinger equations are presented to illustrate the effectiveness of our proposed criteria.

Time Decay Rate of Solutions Toward the Viscous Shock Waves under Periodic Perturbations for the Scalar Conservation Law with Nonlinear Viscosity

Time Decay Rate of Solutions Toward the Viscous Shock Waves under Periodic Perturbations for the Scalar Conservation Law with Nonlinear Viscosity

Global Existence and Time-decay Rates of Solutions to 2D Magneto-micropolar Fluid Equations with Partial Viscosity

Global Existence and Time-decay Rates of Solutions to 2D Magneto-micropolar Fluid Equations with Partial Viscosity

Time decay rate of solutions toward the viscous shock waves to the Cauchy problem for the scalar conservation law with nonlinear viscosity and discontinuous initial data

In this paper, it is shown that the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear viscosity and different far field states, exists globally in time. Especially, this existence holds with discontinuous initial data. Furthermore, when the corresponding Riemann problem for the hyperbolic part has a single shock wave solution, it is proved that the weak solution converges toward the viscous shock wave as time goes to infinity, and the decay rate is also obtained. The proof is given by a technical energy method.

Spectrum Analysis for the Vlasov–Poisson–Boltzmann System

By identifying a norm capturing the effect of the forcing governed by the Poisson equation, we give a detailed spectrum analysis on the linearized Vlasov–Poisson–Boltzmann system around a global Maxwellian. It is shown that the electric field governed by the self-consistent Poisson equation plays a key role in the analysis so that the spectrum structure is genuinely different from the well-known one of the Boltzmann equation. Based on this, we give the optimal time decay rates of solutions to the equilibrium.

The optimal decay rates of classical solutions to the 3D compressible Navier‐Stokes equations

AbstractIn this paper, we revisit the optimal time decay rates of classical solutions to the 3D compressible Navier‐Stokes equations. Based on a global priori estimate, the optimal time decay rates of the solution and its first order derivative in L2‐norm are obtained if ‐norm of the initial perturbation around a constant state is small enough and is bounded for . Compared with the previous works by Hai‐Liang Li and Ting Zhang (Math Meth Appl Sci 34:670‐682, 2011), we remove the smallness of and our condition involves only the low frequencies of the data in .

Optimal time decay rates of solutions for the 2D generalized magneto‐micropolar equations

This study is concerned with the optimal time rates of weak solutions for the 2D magneto‐micropolar equations with only micro‐rotational dissipation and magnetic diffusion. Due to some new observations, we obtain the optimal time decay rates of weak solutions and with p ∈ [2, ∞).

Optimal time decay rates for the compressible Navier-Stokes system with and without Yukawa-type potential

We consider the time decay rates of smooth solutions to the Cauchy problem for the compressible Navier-Stokes system with and without a Yukawa-type potential. We prove the existence and uniqueness of global solutions by the standard energy method under small initial data assumptions. Furthermore, if the initial data belong to \(L^1(\mathbb R^3)\), we establish the optimal time decay rates of the solution as well as its higher-order spatial derivatives. In particular, we obtain the optimal decay rates of the highest-order spatial derivatives of the velocity. Finally, we derive the lower bound time decay rates for the solution and its spacial derivatives. For more information see https://ejde.math.txstate.edu/Volumes/2020/102/abstr.html

On the well-posedness and decay characterization of solutions for incompressible electron inertial Hall-MHD equations

In this paper, by using the properties of decay character and the Fourier splitting method, we establish the time decay rates of solutions for three-dimensional incompressible electron inertial Hall-MHD equations. In particular, the optimal decay rates of the mixed space-time derivatives of the solution are obtained.

Asymptotic profiles of solutions to the generalized double dispersion equation with structural damping

In this paper, we study the Cauchy problem for the generalized double dispersion equation with structural damping. The equation behaves as the usual diffusion phenomenon over the low frequency domain, while it admits a feature of regularity-loss on the high frequency part. The feature of regularity-loss leads to the weakly dissipative property of the equation. To overcome the weakly dissipative property, the time-weighted energy is introduced, and extra regularity on the initial data is required. Under suitable conditions on the initial data and space dimensions, we prove the global existence and time-decay rates of solutions. The proof is based on the spectral analysis for the solution operators, time-weighted energy, and the contraction mapping theorem. Moreover, we also establish the asymptotic profiles of global solutions involving the nonlinear term for n ≥ 3, ν∈(0,12) and n ≥ 4, ν∈[12,1), respectively.

Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain

In this paper we are concerned with a Boussinesq system for small-amplitude long waves arising in nonlinear dispersive media. Considerations will be given for the global well-posedness and the time decay rates of solutions when the model is posed on a periodic domain and a general class of damping operator acts in each equation. By means of spectral analysis and Fourier expansion, we prove that the solutions of the linearized system decay uniformly or not to zero, depending on the parameters of the damping operators. In the uniform decay case, the result is extended for the full system.

Time decay rate of solutions to the tropical climate model equations in ℝ n

This paper is devoted to study the time decay of tropical climate model. We first establish the optimal decay rates in of the weak solutions with large initial data in and obtain the existence and uniqueness of global smooth solutions with small initial data. In addition, taking advantage of Fourier splitting method, we get the decay rates in for higher order derivatives of the smooth solutions with small initial data.

Optimal Decay Rate for the Compressible Flow of Liquid Crystals in $$L^p$$ L p Type Critical Spaces

The purpose of this paper is to establish the optimal time decay rate of solutions for the compressible flow of nematic liquid crystals in $$L^p$$ type critical framework and any dimension $$N\ge 2$$ . Concretely, if the low frequencies of the initial data are in e.g. $$L^{p/2}(\mathbb {R}^N)$$ , we could obtain that the $$L^p$$ norm of the critical global solution $$(\rho -1,\mathbf{u})$$ decays like $$t^{-\frac{N}{p}+ \frac{N}{4}}$$ , while the $$L^p$$ norm of $$\mathbf{d}-\hat{\mathbf{d}}$$ decays like $$t^{-\frac{N}{2p}}$$ for $$t\rightarrow \infty $$ , exactly as shown in the work by Gao et al. (J Differ Equ 261:2334–2383, 2016) when $$p=2$$ and $$N=3$$ , for solutions with high Sobolev regularity. As a by-product, we derive an accurate description of the time decay rates, not only for Lebesgue spaces, but also for a family of Besov norms with negative or nonnegative regularity exponents, which improves the decay results in high Sobolev regularity. The main tools used here are the Littlewood–Paley theory and refined time weighted inequalities in Fourier space.

Well-posedness and decay of solutions for three-dimensional generalized Navier–Stokes equations

In this paper, we study the generalized incompressible Navier–Stokes equations in R3. Based on the energy estimates and regularization of the initial data with the heat semigroup, we prove the well-posedness of solutions in H5−4α2(R3) provided that the H5−4α2(R3)-norm of initial data is sufficiently small. In addition, in contrast to the generalized heat equation, the upper bound of the time decay rate of solutions to the generalized Navier–Stokes equations is also established.

Asymptotic Behavior of Solutions to a New Hall-MHD System

In this paper, by using Fourier splitting method and the properties of decay character $r^{*}$ , we consider the algebraic time decay rate of solutions to a new Hall-MHD system in Sobolev space $H^{m}(\mathbb{R} ^{3})\times H^{m+1}(\mathbb{R}^{3})$ for $m\geq 0$ .

Global Existence and Optimal Decay Rates for Three-Dimensional Compressible Viscoelastic Flows System with Damping

In this paper, we consider the large time behavior of the strong solutions to the three dimensional compressible viscoelastic flows with damping. Based on the energy method and spectral analysis, we analyze the influences of the damping on the global existence and decay rates of compressible viscoelastic flows under some small assumptions in H3-framework. Compared with the time decay rates of solutions to the compressible viscoelastic flows in [1], our results imply that the friction of the damping is stronger than the dissipation effect of the viscosities.

Decay of the 3D viscous liquid-gas two-phase flow model with damping

We establish the optimal Lp − L2(1 ≤ p &amp;lt; 6/5) time decay rates of the solution to the Cauchy problem for the 3D viscous liquid-gas two-phase flow model with damping and analyse the influences of the damping on the qualitative behaviors of solution. It is observed that the fraction effect of the damping affects the dispersion of fluids and enhances the time decay rate of solution. Our method of proof consists of Hodge decomposition technique, Lp − L2 estimates for the linearized equations, and delicate energy estimates.

Time decay rate of solutions to the hyperbolic mhd equations in [formula omitted

In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics (MHD) equations in ℝ3. Then we establish that the solutions with initial data belonging to Hm(ℝ3)∩L1(ℝ3) have the following time decay rate: ‖∇mu(x,t)‖2+‖∇mb(x,t)‖2+‖∇m+1u(x,t)‖2+‖∇m+1b(x,t)‖2≤c(1+t)-3/2-m for large t, where m = 0, 1.

Time decay rates for the equations of the compressible heat-conductive flow through porous media

We consider the time decay rates of smooth solutions to the Cauchy problem for the equations of the compressible heat-conductive flow through porous media. We prove the global existence and uniqueness of the solutions by the standard energy method. Moreover, we establish the optimal decay rates of the solution as well as its higher-order spatial derivatives. And the damping effect on the time decay rates of the solution is studied in detail.