Asymptotic Behavior Of Strong Solutions Research Articles

On the Cauchy problem of 3D nonhomogeneous micropolar fluids with density-dependent viscosity

<p>In this paper, we considered the global well-posedness of strong solutions to the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible micropolar fluids with density-dependent viscosity and vacuum. Based on the energy method, some key a priori exponential decay-in-time rates of strong solutions are obtained. As a result, the existence and large-time asymptotic behavior of strong solutions in the whole space $ \mathbb{R}^3 $ are established, provided that the initial mass is sufficiently small. Note that this result is proven without any compatibility conditions.</p>

The [formula omitted]-asymptotic behavior of strong solutions to the incompressible magneto-hydrodynamic equations in half-spaces

We give the L1-decay estimates on the fractional spatial derivatives of the strong solution to the incompressible magneto-hydrodynamic (MHD) equations in a half-space. Moreover, by using the Stokes solution formula and the decomposition of the nonlinear terms in MHD equations, we derive asymptotic expansion properties of the strong solution in L1(R+n).

Well-posedness and asymptotic behavior of stochastic convective Brinkman–Forchheimer equations perturbed by pure jump noise

This paper is concerned about stochastic convective Brinkman–Forchheimer (SCBF) equations subjected to multiplicative pure jump noise in bounded or periodic domains. Our first goal is to establish the existence of a pathwise unique strong solution satisfying the energy equality (Itô’s formula) to SCBF equations. We resolve the issue of the global solvability of SCBF equations, by using a monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty–Browder technique. The major difficulty is that an Itô’s formula in infinite dimensions is not available for such systems. This difficulty is overcame by the technique of approximating functions using the elements of eigenspaces of the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. Due to some technical difficulties, we discuss the global in time regularity results of such strong solutions in periodic domains only. Once the system is well-posed, we look for the asymptotic behavior of strong solutions. For large effective viscosity, the exponential stability results (in the mean square and pathwise sense) for stationary solutions is established. Moreover, a stabilization result of SCBF equations by using a multiplicative pure jump noise is also obtained. Finally, we prove the existence of a unique ergodic and strongly mixing invariant measure for SCBF equations subject to multiplicative pure jump noise, by using the exponential stability of strong solutions.

Global Well-Posedness and Exponential Stability of 3D Navier–Stokes Equations with Density-Dependent Viscosity and Vacuum in Unbounded Domains

We consider the global existence and large-time asymptotic behavior of strong solutions to the Cauchy problem of the three-dimensional (3D) nonhomogeneous incompressible Navier–Stokes equations with density-dependent viscosity and vacuum. After establishing some key a priori exponential decay-in-time rates of the strong solutions, we obtain both the global existence and exponential stability of strong solutions in the whole three-dimensional space, provided that the initial velocity is suitably small in some homogeneous Sobolev space which may be optimal compared with the case of homogeneous Navier-Stokes equations. Note that this result is proved without any smallness conditions on the initial density which contains vacuum and even has compact support.

Asymptotic behavior of strong solutions of a simplified energy-transport model with general conductivity

In this paper we study a simplified transient energy-transport model in semiconductors with a general conductivity and the Dirichlet boundary conditions on an interval. By using a new iterative scheme, we prove the global existence and uniqueness of strong solutions provided that the variation of the temperature is small. Also, the existence and stability of stationary solutions are proved if the temperature is large.

On the Cauchy problem of 3D nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and vacuum

This paper concerns the Cauchy problem of the three-dimensional nonhomogeneous incompressible magnetohydrodynamic (MHD) equations with density-dependent viscosity and vacuum. We first establish some key a priori algebraic decay-in-time rates of the strong solutions. Then after using these estimates, we also obtain the global existence and large time asymptotic behavior of strong solutions in the whole three-dimensional space, provided that the initial velocity and magnetic field are suitable small in the $$\dot{H}^{\beta }$$ -norm for some $$\beta \in (1/2, 1]$$ . Note that any smallness and compatibility conditions assumed on the initial data are not used in this result. Moreover, the density can contain vacuum states and even have compact support initially.

Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier–Stokes equations with vacuum

We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Navier–Stokes equations with vacuum as far-field density. It is proved that if the initial density decays not too slow at infinity, the 2D Cauchy problem of the density-dependent Navier–Stokes equations on the whole space admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Furthermore, we also obtain the large time decay rates of the spatial gradients of the velocity and the pressure, which are the same as those of the homogeneous case.

Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain

In this paper, we investigate global existence and asymptotic behavior of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain with no-slip boundary. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in $H^2(Ω)$ . Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

On a new two-component $b$-family peakon system with cubic nonlinearity

In this paper, we propose a two-component \begin{document}$b$\end{document} -family system with cubic nonlinearity and peaked solitons (peakons) solutions, which includes the celebrated Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and its two-component extension as special cases. Firstly, we study single peakon and multi-peakon solutions to the system. Then the local well-posedness for the Cauchy problem of the system is discussed. Furthermore, we derive the precise blow-up scenario and global existence for strong solutions to the two-component \begin{document}$b$\end{document} -family system with cubic nonlinearity. Finally, we investigate the asymptotic behaviors of strong solutions at infinity within its lifespan provided the initial data decay exponentially and algebraically.

On a class of fully nonlinear parabolic equations

Abstract We study the homogeneous Dirichlet problem for the fully nonlinear equation u_{t}=|\Delta u|^{m-2}\Delta u-d|u|^{\sigma-2}u+f\quad\text{in ${Q_{T}=\Omega% \times(0,T)}$,} with the parameters {m>1} , {\sigma>1} and {d\geq 0} . At the points where {\Delta u=0} , the equation degenerates if {m>2} , or becomes singular if {m\in(1,2)} . We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as {t\to\infty} . Sufficient conditions for exponential or power decay of {\|\nabla u(t)\|_{2,\Omega}} are derived. It is proved that for certain ranges of the exponents m and σ, every strong solution vanishes in a finite time.

Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum

This paper concerns the Cauchy problem of the nonhomogeneous incompressible Magnetohydrodynamic equations on the whole two-dimensional space with vacuum as far-field density. We establish the global existence and uniqueness of strong solutions to the 2D Cauchy problem provided that the initial density and the initial magnetic field decay not too slow at infinity. In particular, the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Furthermore, we also obtain the large time decay rates of the gradients of velocity, magnetic field and pressure.

Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system

In this paper, we first address the space‐time decay properties for higher‐order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument, and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions for small initial data in some scaling invariant function spaces. The smallness conditions are somehow weaker than those presented by Brandolese and Schonbek. We further investigate the asymptotic profiles and decay properties of these strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.

Global existence and large time asymptotic behavior of strong solutions to the 2-D compressible magnetohydrodynamic equations with vacuum

The authors study the Cauchy problem of the magnetohydrodynamic equations for viscous compressible barotropic flows in two or three spatial dimensions with vacuum as far field density. For two spatial dimensions, we establish the global existence and uniqueness of strong solutions (which may be of possibly large oscillations) provided the smooth initial data are of small total energy, and obtain some a priori decay with rates (in large time) for the pressure, the spatial gradient of both the velocity field and the magnetic field. Moreover, for three spatial dimensions case, some similar decay rates are also obtained.

Asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term

This paper deals with the asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term |u|β−1u(β≥3). First, we establish an upper bound for the difference between the solution of our equation and the heat equation in L2 space. Then, we optimize the upper bound of decay for the solutions and obtain their algebraic lower bound by using Fourier Splitting method.

Determining nodes for semilinear parabolic equations

We are concerned with the determination of the asymptotic behavior of strong solutions to the initial-boundary value problem for general semilinear parabolic equations by the asymptotic behavior of these strong solutions on a finite set. More precisely, if the asymptotic behavior of the strong solution is known on a suitable finite set which is called determining nodes, then the asymptotic behavior of the strong solution itself is entirely determined. We prove the above property by the energy method.

On the primitive equations of large-scale ocean with random boundary

This paper is concerned with global well posedness and the existence of attractors for the three-dimensional viscous primitive equations, describing large-scale oceanic motion under random wind stress. We prove global well posedness of the primitive equations with white-noise boundary conditions. Moreover, by studying the asymptotic behavior of strong solutions, we obtain the existence of random attractors for the corresponding random dynamical system.

3D Stochastic Primitive Equations of the Large-Scale Ocean: Global Well-Posedness and Attractors

In this paper, we consider the global well-posedness and long-time dynamics for the three-dimensional viscous primitive equations describing the large-scale oceanic motion under a random forcing, which is an additive white in time noise. We firstly prove the existence and uniqueness of global strong solutions to the initial boundary value problem for the stochastic primitive equations. Subsequently, by studying the asymptotic behavior of strong solutions, we obtain the existence of random attractors for the corresponding random dynamical system.

Asymptotic behavior of strong solutions for nonlinear parabolic equations with critical Sobolev exponent

In this paper, we discuss the asymptotic behavior of solutions of nonlinear parabolic equations in ${\mathbf{R}}^N$ involving critical Sobolev exponent. For the semilinear and subcritical problem, it is well-known that the solution which intersects with the ``stable set'' must be a global one and the solution which enters the "unstable set"feodory should blow up in finite time. But in the critical case, it is not clear that the same result holds or not. In this paper, we show that the same result holds also in the critical case. The proof of our main result requires the method different from that for the subcritical problem and is based on the direct analysis of $L^\infty$-norm of solutions with the aid of the blow up argument and the concentration compactness type argument.