A Note on Time-Decay Estimates for the Compressible Navier–Stokes Equations

In the recent work, we have developed a decay framework in general L~p critical spaces and established optimal time-decay estimates for barotropic compressible Navier–Stokes equations. Those decay rates of L~q-L~r type of the solution and its derivatives are available in the critical regularity framework, which were exactly firstly observed by Matsumura Nishida, and subsequently generalized by Ponce for solutions with high Sobolev regularity. We would like to mention that our approach is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. In this paper, a new observation is involved in the high frequency, which enables us to improve decay exponents for the high frequencies of solutions.     

Existence and asymptotic behavior for an incompressible Newtonian flow with intrinsic degree of freedom

We consider the Cauchy problem of a mathematical model for an incompressible, homogeneous, Newtonian fluid taking into account internal degrees of freedom. We first show that there exists a unique global strong solution when the given initial data are small in some sense. Then, we deduce the optimal decay rates for velocity vector in L² ? norm and L^p ? norm for p > n. These decay estimates depend only on the spatial dimension and the decay properties of the heat solution with the same data. Copyright © 2013 John Wiley & Sons, Ltd.     

GLOBAL EXISTENCE AND LONG-TIME BEHAVIOR FOR THE STRONG SOLUTIONS IN H2 TO THE 3D COMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOWS

In this paper, we investigate the global existence and long time behavior of strong solutions for compressible nematic liquid crystal flows in three-dimensional whole space. The global existence of strong solutions is obtained by the standard energy method under the condition that the initial datas are close to the constant equilibrium state in H²-framework. If the initial datas in L¹-norm are finite additionally, the optimal time decay rates of strong solutions are established. With the help of Fourier splitting method, one also establishes optimal time decay rates for the higher order spatial derivatives of director.     

Existence and stability of planar diffusion waves for 2-D Euler equations with damping

To study the non-linear stability of a non-trivial profile for a multi-dimensional systems of gas dynamics, the combination of the Green function on estimating the lower order derivatives and the energy method for the higher order derivatives is shown to be not only useful but sometimes maybe also essential. In this paper, we study the stability of a planar diffusion wave for the isentropic Euler equations with damping in two-dimensional space. By introducing an approximate Green function for the linearized equations around the planar diffusion wave and by applying the energy method, we prove the global existence and the L₂ convergence rate of the solution when the initial data is a small perturbation of the planar diffusion wave. The decay rates of the perturbation and its lower order spatial derivatives obtained are optimal in the L₂ norm. Furthermore, the constructed approximate Green function in this paper can be used for the pointwise and the Lp estimates of the solutions concerned. In fact, the approach by combining of the Green function and energy method can be applied to other system especially when the derivatives of the coefficients in the system have certain time decay properties.     

A priori error analysis of a discontinuous Galerkin approximation for a kind of compressible miscible displacement problems

A kind of compressible miscible displacement problems which include molecular diffusion and dispersion in porous media are investigated.A symmetric interior penalty discontinuous Galerkin (SIPG) method is applied to the coupled system of flow and transport.Using the induction hypotheses instead of the cut-off operator and the interpolation projection properties,a priori hp error estimates are presented.The error bounds in L2(H1) norm for concentration and in L∞(L2) norm for velocity are optimal in h and suboptimal in p with a loss of power 1/2.     

Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in R³. We prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H³-framework. Moreover, if additionally the initial data belong to L^p with , the optimal convergence rates of the solutions in L^q-norm with 2≤q≤6 and its spatial derivatives in L²-norm are obtained.     

Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

We investigate a model arising from biology, which is a hyperbolic- parabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the Hs ∩ Ll-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L2 decay rate of the solution and the almost optimal L2 decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.     

Large time behavior of solutions to 3D compressible Navier-Stokes-Poisson system

We consider the three-dimensional compressible Navier-Stokes-Poisson system where the electric field of the internal electrostatic potential force is governed by the self-consistent Poisson equation.If the Fourier modes of the initial data are degenerate at the low frequency or the initial data decay fast at spatial infinity,we show that the density converges to its equilibrium state at the L 2-rate (1+t)(-7/4) or L ∞-rate (1+t)(-5/2),and the momentum decays at the L 2-rate (1+t)(-5/4) or L ∞-rate (1+t)(-2).These convergence rates are shown to be optimal for the compressible Navier-Stokes-Poisson system.     

Asymptotic behavior of solutions to the compressible nematic liquid crystal system in ℝ3

In this paper, we study a nematic liquid crystals system in three-dimensional whole space ?³ and obtain the time decay rates of the higher-order spatial derivatives of the solution by the method of spectral analysis and energy estimates if the initial data belongs to L¹?³ additionally.     

Global existence and convergence rates of smooth solutions for the full compressible MHD equations

In this paper, we consider the global smooth solutions and their decay for the full compressible magnetohydrodynamic equations in R ³. We prove the global existence of smooth solutions near the constant state in Sobolev norms by energy method and show the convergence rates of the L ^p -norm of these solutions to the constant state when the L ^q -norm of the perturbation is bounded.     

FBSDEs with time delayed generators: L-solutions, differentiability, representation formulas and path regularity

We extend the work of Delong and Imkeller (2010) [6] and [7] concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general L^p-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in L^p. We introduce decoupled systems of SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic L²-path regularity to delay FBSDEs.     

A scale-invariant Klein–Gordon model with time-dependent potential

The goal of the present paper is to derive statements about energy estimates as well as L ^p ?L ^q decay estimates for a Klein?CGordon model with a particular time-dependent mass. The study of this special case of a scale-invariant model is an important step within a systematic investigation of Klein?CGordon models with time-dependent mass.     

On functions of bounded p-variation

We obtain estimates of the total p-variation (1<p<∞) and other related functionals for a periodic function f∈Lp[0,1] in terms of its Lp-modulus of continuity ωp(f;δ). These estimates are sharp for any rate of the decay of ωp(f;δ). Moreover, the constant coefficients in them depend on parameters in an optimal way.     

A superconvergent local discontinuous Galerkin method for nonlinear two-point boundary-value problems

In this paper, we present and analyze a superconvergent and high order accurate local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form u ^″ = f (t, u), which arise in a wide variety of engineering applications. We prove the L ² stability of the LDG scheme and optimal L ² error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p +?1, when piecewise polynomials of degree at most p are used. Our numerical experiments demonstrate optimal rates of convergence. Moreover, we show that the derivatives of the LDG solutions are superconvergent with order p +?1 toward the derivatives of Gausss-Radau projections of the exact solutions. Finally, we prove that the LDG solutions are superconvergent with order p +?3/2 toward Gauss-Radau projections of the exact solutions. Our computational results indicate that the observed numerical superconvergence rate is p +?2. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p ≥?1 and for the periodic, Dirichlet, and mixed boundary conditions. All proofs are valid under the hypotheses of the existence and uniqueness theorem for BVPs. Several numerical results are presented to validate the theoretical results.     

Decay of a model system of radiating gas

This paper concerns the Cauchy problem of a model system to the radiating gas in . Large time behaviors of classical solutions to the Cauchy problem are studied without needing the smallness assumption of initial perturbation in L¹‐norm. We obtain the optimal H^N‐norm time‐decay rates of the solutions in with 1 ≤ n ≤ 4 by applying the Fourier splitting method introduced by Schonbek (1980) with a slight modification and an energy method. Furthermore, when initial perturbation is bounded in L^p‐norm (p ∈ (1,2]), optimal L^p‐ L² decay estimates of the derivatives of solutions are shown. Copyright © 2013 John Wiley & Sons, Ltd.     

On the wave equation with a large rough potential

We prove an optimal dispersive L^∞ decay estimate for a three-dimensional wave equation perturbed with a large nonsmooth potential belonging to a particular Kato class. The proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator.     

Pointwise estimates of solutions for the multi-dimensional bipolar Euler–Poisson system

In the paper, we consider a multi-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equations. By making a new analysis on Green’s functions for the Euler system with damping and the Euler–Poisson system with damping, we obtain the pointwise estimates of the solution for the multi-dimensions bipolar Euler–Poisson system. As a by-product, we extend decay rates of the densities \({\rho_i(i=1,2)}\) in the usual L²-norm to the L^p-norm with \({p\geq1}\) and the time-decay rates of the momentums m_i(i = 1,2) in the L²-norm to the L^p-norm with p > 1 and all of the decay rates here are optimal.     

On upper and lower bounds of higher order derivatives for solutions to the 2D micropolar fluid equations

The present paper is concerned with asymptotic behaviours of the solutions to the micropolar fluid motion equations in R². Upper and lower bounds are derived for the L² decay rates of higher order derivatives of solutions to the micropolar fluid flows. The findings are mainly based on the basic estimates of the linearized micropolar fluid motion equations and generalized Gronwall type argument.     

Optimal decay rates for the system of elastic waves in R n with structural damping

We consider the Cauchy problem in R ⁿ for the system of elastic waves with structural damping. We derive (almost) optimal decay rates in time for the L ²-norm and the total energy which improves previous results for this system. To derive the estimates for elastic waves, we employ an improvement in a method in the Fourier space, which was developed in our previous works. Our estimates came from those for a generalized energy of α-order in the Fourier space.