The blow-up rates for systems of heat equations with nonlinear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions. The upper and lower bounds of blow-up rate are obtained.     

Gevrey Class Regularity and Exponential Decay Property for Navier-Stokes-α Equations

The Navier-Stokes-α equations subject to the periodic boundary conditions are considered.An-alyticity in time for a class of solutions taking values in a Gevrey class of functions is proven.Exponentialdecay of the spatial Fourier spectrum for the analytic solutions and the lower bounds on the rate defined by theexponential decay are also obtained.     

Blowup behaviors for diffusion system coupled through nonlinear boundary conditions in a half space

This paper deals with the blowup estimates near the blowup time for the system of heat equations in a half space coupled through nonlinear boundary conditions. The upper and lower bounds of blowup rate are established. The uniqueness and nonunique-ness results for the system with vanishing initial value are given.     

ASYMPTOTIC BEHAVIOR OF THE NONLINEAR PARABOLIC EQUATIONS

This paper is concerned with the large time behavior for solutions of the nonlinear parabolic equations in whole spaces R^n. The spectral decomposition methods of Laplace operator are applied and it is proved that if the initial data u0∈ L^2 ∩ L^r for 1 ≤ r ≤ 2, then the solutions decay in L^2 norm at t^-n/2(1/r-1/2). The decay rates are optimal in the sense that they coincide with the decay rates of the solutions to the heat equations with the same initial data.     

PROPERTIES OF SOLUTIONS OF n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Consider the n-dimensional incompressible Navier-Stokes equations ?/(?t)u-α△u +(u · ?)u + ?p = f(x, t), ? · u = 0, ? · f = 0,u(x, 0) = u0(x), ? · u0= 0.There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations.· Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?· Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?· Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?· If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?· Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order m of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.     

The high-order SPDEs driven by multi-parameter fractional noises

The existence and uniqueness of the solutions are proved for a class of fourth-order stochastic heat equations driven by multi-parameter fractional noises. Furthermore the regularity of the solutions is studied for the stochastic equations and the existence of the density of the law of the solution is obtained.     

On the decay of higher order derivatives of solutions to Ladyzhenskaya model for incompressible viscous flows

This article concerns large time behavior of Ladyzhenskaya model for incompressible viscous flows in R~3.Based on linear L~P-L~q estimates,the auxiliary decay properties of the solutions and generalized Gronwall type arguments,some optimal upper and lower bounds for the decay of higher order derivatives of solutions are derived without assuming any decay properties of solutions and using Fourier splitting technology.     

Some remarks on planar Boussinesq equations

The main purpose of this paper is to prove the well-posedness of the two-dimensional Boussinesq equations when the initial vorticity ω 0 ∈L1 (R 2 ) (or the finite Radon measure space). Using the stream function form of the equations and the Schauder fixed-point theorem to get the new proof of these results, we get that when the initial vorticity is smooth, there exists a unique classical solutions for the Cauchy problem of the two dimensional Boussinesq equations.     

Global Attractors and Determining Modes for the 3D Navier-Stokes-Voight Equations

The authors investigate the long-term dynamics of the three-dimensional Navier- Stokes-Voight model of viscoelastic incompressible fluid. Specifically, upper bounds for the number of determining modes are derived for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view the authors consider the Navier-Stokes-Voight model as a non-viscous （inviscid） regularization of the three-dimensional Navier-Stokes equations. Furthermore, it is also shown that the weak solutions of the Navier-Stokes- Voight equations converge, in the appropriate norm, to the weak solutions of the inviscid simplified Bardina model, as the viscosity coefficient v →0.     

Partial compactness for Landau-Lifshitz maxwell equation in two-dimension

We study the partial regularity of weak solutions to the 2-dimensional LandauLifshitz equations coupled with time dependent Maxwell equations by Ginzburg-Landau type approximation. Outside an energy concentration set of locally finite 2-dimensional parabolic Hausdorff measure, we prove the uniform local C ∞ bounds for the approaching solutions and then extract a subsequence converging to a global weak solution of the Landau-Lifshitz-Maxwell equations which are smooth away from finitely many points.     

Lp estimates of solutions of some partial differential equations

The author is concerned with the long time asymptotic behaviors of the global weak solutions of some nonlinear evolution equations. First of all, he derives some uniform L¹ and L^∞ upper bounds for the solutions, under some mild conditions. Then, by applying the well-known Fourier splitting method and the L¹ estimates, he asserts the L² decay estimates of the solutions. The rates of decay are sharp in the sense that the integral of the initial data over $\text{R}$ is nonzero.     

Global dynamics of the Boussinesq‐Burgers system with large initial data

In this paper, we study the global existence and asymptotic behavior of the Boussinesq‐Burgers system subject to the Dirichlet boundary conditions. Based on the L^p(p > 2) estimates of the solution, which are different from the standard L²‐based energy methods, we show that the classical solutions exist globally and converge to their boundary data at an exponential decay rate as time goes to infinity for large initial data. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Lower Bounds and Nonuniform Time Discretization for Approximation of Stochastic Heat Equations

We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0, 1[^d. The error of an algorithm is defined in L₂-sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W. For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of nonuniform time discretizations is crucial.     

On the decay and stability of global solutions to the 3‐D inhomogeneous Navier‐Stokes equations

In this paper, we investigate the large‐time decay and stability to any given global smooth solutions of the 3‐D incompressible inhomogeneous Navier‐Stokes equations. In particular, we prove that given any global smooth solution (a,u) of (1.2), the velocity field u decays to 0 with an explicit rate, which coincides with the L² norm decay for the weak solutions of the 3‐D classical Navier‐Stokes system [26,29] as t goes to ∞. Moreover, a small perturbation to the initial data of (a,u) still generates a unique global smooth solution to (1.2), and this solution keeps close to the reference solution (a,u) for t > 0. We should point out that the main results in this paper work for large solutions of (1.2). © 2010 Wiley Periodicals, Inc.     

Some nonstandard problems in generalized heat conduction

The Maxwell-Cattaneo system of equations for generalized heat conduction is considered where the temperature and heat flux, respectively, are subject to auxiliary conditions which prescribe a combination of their values initially and at a later time. By means of differential inequalities, L₂ exponential decay bounds for the temperature and heat flux are determined in terms of data for a range of values of the parameter in the nonstandard auxiliary condition. Decay bounds are also obtained in two related problems.     

Some nonstandard problems in generalized heat conduction

The Maxwell-Cattaneo system of equations for generalized heat conduction is considered where the temperature and heat flux, respectively, are subject to auxiliary conditions which prescribe a combination of their values initially and at a later time. By means of differential inequalities, L₂ exponential decay bounds for the temperature and heat flux are determined in terms of data for a range of values of the parameter in the nonstandard auxiliary condition. Decay bounds are also obtained in two related problems. Received: July 14, 2003     

Robin Approximation of Dirichlet Boundary Value Problems

This paper describes the rate of convergence of solutions of Robin boundary value problems of an elliptic equation to the solution of a Dirichlet problem as a boundary parameter decreases to zero. The results are found using representations for solutions of the equations in terms of Steklov eigenfunctions. Particular interest is in the case where the Dirichlet data is only in L²(?Ω,dσ). Various approximation bounds are obtained and the rate of convergence of the Robin approximations in the H¹ and L² norms are shown to have convergence rates that depend on the regularity of the Dirichlet data.     

Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.