Pointwise Estimates Of Solutions Research Articles

Long-time behavior of a point mass in a one-dimensional viscous compressible fluid and pointwise estimates of solutions

We consider the motion of a point mass in a one-dimensional viscous compressible barotropic fluid. The fluid–point mass system is governed by the barotropic compressible Navier–Stokes equations and Newton's equation of motion. Our main result concerns the long-time behavior of the fluid and the point mass. Pointwise convergence estimates of the volume ratio and the velocity of the fluid to their equilibrium values are shown, and as a corollary, it is proved that the velocity V(t) of the point mass satisfies a decay estimate |V(t)|=O(t−3/2). The rate −3/2 is optimal and is essentially related to the compressibility and the nonlinearity. The main tool used in the proof is the pointwise estimates of Green's function. It turns out that the understanding of the time-decay properties of the transmitted and reflected waves at the point mass plays an essential role in the proof.

Pointwise Estimates of Solutions for the Viscous Cahn-Hilliard Equation with Inertial Term

In this paper, we study the pointwise estimates of solutions to the viscous Cahn-Hilliard equation with the inertial term in multidimensions. We use Green’s function method. Our approach is based on a detailed analysis on the Green’s function of the linear system. And we get the solution’s Lp convergence rate.

Pointwise estimates of solutions for the nonlinear viscous wave equation in even dimensions

In this article, we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions (n ≥ 4). We use the Green’s function method. Our approach is on the basis of the detailed analysis of the Green’s function of the linearized system. We show that the decay rates of the solution for the same problem are different in even dimensions and odd dimensions. It is shown that the solution exhibits a generalized Huygens principle.

Weighted Parabolic Aleksandrov Estimates: PDE and Stochastic Versions

We prove several pointwise estimates for solutions of linear parabolic equations with measurable coefficients in smooth cylinders through the weighted Ld+1-norm of the free term. The weights allow the free term to blow up near the lateral boundary. These are the first such results for parabolic equations with measurable coefficients. We also present the corresponding weighted estimates for occupation times of diffusion processes.

Boundary behavior of blowup solutions for a heat equation with a nonlinear boundary condition

This paper is concerned with positive blowup solutions for the heat equation with a nonlinear boundary condition. The result in this paper provides a precise description of the blowup profile. A key part of the proof is a pointwise estimate of solutions near the blowup points. This estimate is obtained by the study of a stationary problem of some rescaled equation. Furthermore we provide uniform upper bounds of positive blowup solutions by the Liouville type theorem for a rescaled equation in similarity variables.

Global Existence and Pointwise Estimates of Solutions to Generalized Kuramoto-Sivashinsky System in Multi-Dimensions

The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multi-dimensions (n ≥ 3) is considered. Based on Green’s function method, some ingenious energy estimates are given. Then the global existence and pointwise convergence rates of the classical solutions are established. Furthermore, the Lp convergence rate of the solution is obtained.

Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms

We are concerned with the pointwise estimates of solutions to the scalar conservation law with a nonlocal dissipative term for arbitrary large initial data. Based on the Green's function method, time-frequency decomposition method as well as the classical energy estimates, pointwise estimates and the optimal decay rates are established in this paper. We emphasize that the decay rate is independent of the index s in the nonlocal dissipative term. This phenomenon is also coincident with the fact that the decay rate is determined by the low frequency part of the solution no matter the initial data is small or large.

Pointwise estimates of solutions to the double-phase elliptic equations

With the help of nonlinear Wolf potentials, we derive the pointwise estimates for the weak solutions to inhomogeneous quasilinear double-phase elliptic equations of the divergence type.

Global existence and pointwise estimates of solutions for the generalized sixth-order Boussinesq equation

Global existence and pointwise estimates of solutions for the generalized sixth-order Boussinesq equation

The pointwise estimates of solutions for a nonlinear convection diffusion reaction equation

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension. First, the pointwise estimates of solutions are obtained, furthermore, we obtain the optimal Lp, 1 ≤ p ≤ + ∞, convergence rate of solutions for small initial data. Then we establish the local existence of solutions, the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data. Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.

The pointwise estimates of solutions to the Cauchy problem of a chemotaxis model

This paper deals with an attraction-repulsion chemotaxis model (ARC) in multi-dimensions. By Duhamel’s principle, the implicit expression of the solution to (ARC) is given. With the method of Green’s function, the authors obtain the pointwise estimates of solutions to the Cauchy problem (ARC) for small initial data, which yield the Ws,p (1 ≤ p ≤ ∞) decay properties of solutions.

Pointwise estimate of solutions to the three-dimensional compressible magnetohydrodynamic equations

This paper is devoted to the pointwise estimate of solutions for the initial value problem to the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the detailed analysis of the Green function of the linearized system, we obtain the pointwise estimates of smooth solutions when the initial data is sufficiently small with the algebraic decay to the constant equilibrium. As the by-product, we also show the corresponding Lp-estimates of the smooth solutions.

On the convergence of the conditional gradient method as applied to the optimization of an elliptic equation

The optimal control of a second-order semilinear elliptic diffusion-reaction equation is considered. Sufficient conditions for the convergence of the conditional gradient method are obtained without using assumptions (traditional for optimization theory) that ensure the Lipschitz continuity of the objective functional derivative. The total (over the entire set of admissible controls) preservation of solvability, a pointwise estimate of solutions, and the uniqueness of a solution to the homogeneous Dirichlet problem for a controlled elliptic equation are proved as preliminary results, which are of interest on their own.

Global existence and pointwise estimates of solutions to generalized Benjamin-Bona-Mahony equations in multi dimensions

This paper is concerned with the global existence and pointwise estimates of solutions to the generalized Benjamin-Bona-Mahony equations in all space dimensions. By using the energy method, Fourier analysis and pseudo-differential operators, the global existence and pointwise convergence rates of the solution are obtained. The decay rate is the same as that of the heat equation and one can see that the solution propagates along the characteristic line.

Pointwise estimates of solution to compressible nematic liquid crystal flow in odd dimensions

Pointwise estimates of solution to compressible nematic liquid crystal flow in odd dimensions

Pointwise Estimates for Solutions to a System of Radiating Gas

In this paper we focus on the initial value problem of a hyperbolic-elliptic coupled system in multi-dimensional space of a radiating gas. By using the method of Green function combined with Fourier analysis, we obtain the pointwise decay estimates of solutions to the problem.

Asymptotic behavior of solutions to conservation laws with diffusion-type terms of regularity-gain and regularity-loss

In this paper, we study asymptotic behaviors of solutions to the Cauchy problem of nonlinear conservation laws with a diffusion-type source term related to an index s∈R. For s≤1 and s>1, the diffusion-type term takes on a characteristic of regularity-gain and regularity-loss on the high frequency domain, respectively. By combining the Green function method with the energy method, we overcome the weakly dissipative structure of the equation for the case of s>1 and obtain the global existence and optimal Lp-norm time-decay rates of solutions. In the case of regularity-gain, pointwise estimates of solutions are shown by using the refined analysis on the Green function.

On improved estimates for parabolic equations with double degeneracy

We obtain improved pointwise estimates for solutions to nonlinear parabolic equations with double degeneracy. These estimates differ from the classical estimates known for the linear case in that the supremum of the modulus of a solution is estimated by the sum of two powers of the integral norm of the solution.

Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions

The global existence and pointwise estimates of the Cauchy problem for the Euler-Poisson equation with damping in multi-dimensions are considered. Based on the analysis of Green function, and using the special structure of the system together with weighted energy method, we obtain the global existence of the classical solution. What's more important, is that we derive a detailed, pointwise description of asymptotic behavior of the solutions of the Cauchy problem. Then we obtain the optimal $L^p(R^n)\ (p>\frac{n}{n-1})$ convergence rate of the solutions.

Nonlinear parabolic problems in perforated domains

The behavior of the remainder term of the asymptotic expansion for solutions of a quasi-linear parabolic Cauchy-Dirichlet problem in a sequence of domains with fine-granulated boundary is studied. By using a modification of the asymptotic expansion and new pointwise estimates of solutions of the model problem, the uniform convergence of the remainder term to zero is proved.