Higher Order Derivatives Of Solutions Research Articles

Decay of higher order derivatives for Lp solutions to the compressible fluid model of Korteweg type

As a follow-up study of [31], we present a derivation of upper bounds for the decay of arbitrary higher order derivatives of solutions to the compressible Navier-Stokes-Korteweg system in the Lp critical Besov space. The approach, based on Gevrey regularizing estimates, is to establish uniform bounds on the growth of the radius of analyticity of the solution in negative Besov norms, which may be applicable to a wide range of dissipative systems.

Lower bound of decay rate for higher order derivatives of solution to the compressible quantum magnetohydrodynamic model

The lower bounds of decay rates for global solution to the compressible viscous quantum magnetohydrodynamic model in three‐dimensional whole space under the H5 × H4 × H4 framework are investigated in this paper. We first show that the lower bound of decay rate for the solution converging to constant equilibrium state (1, 0, 0) in L2‐norm is when the initial data satisfy some low‐frequency assumption. Moreover, we prove that the lower bound of decay rate of k(k ∈ [1, 3]) order spatial derivative for the solution converging to constant equilibrium state (1, 0, 0) in L2‐norm is . Then, we show that the lower bound of decay rate for the time derivatives of density and velocity is , but the lower bound of decay rate for the time derivative of magnetic field converging to zero in L2‐norm is .

Global axisymmetric classical solutions of full compressible magnetohydrodynamic equations with vacuum free boundary and large initial data

In this paper, the global existence of the classical solution to the vacuum free boundary problem of full compressible magnetohydrodynamic equations with large initial data and axial symmetry is studied. The solutions to the system (1.6)–(1.8) are in the class of radius-dependent solutions, i.e., independent of the axial variable and the angular variable. In particular, the expanding rate of the moving boundary is obtained. The main difficulty of this problem lies in the strong coupling of the magnetic field, velocity, temperature and the degenerate density near the free boundary. We overcome the obstacle by establishing the lower bound of the temperature by using different Lagrangian coordinates, and deriving the uniform-in-time upper and lower bounds of the Lagrangian deformation variable rx by weighted estimates, and also the uniform-in-time weighted estimates of the higher order derivatives of solutions by delicate analysis.

Smoothness in the Dini Space of a Single Layer Potential for a Parabolic System in the Plane

We prove estimates in the Dini space for the second order spatial derivative of a vector parabolic single layer potential. We obtain estimates for higher order derivatives of solutions to initial-boundary-value problems for one-dimensional parabolic systems with Dini-continuous coefficients in a curvilinear domain with nonsmooth lateral boundary.

Regularizing rate estimates for the 3-D incompressible micropolar fluid system in critical Besov spaces

ABSTRACTIn this paper, we are concerned with regularizing rates of the higher order derivatives of solutions for the initial value problem of a three-dimensional incompressible micropolar fluid system with initial data for . More precisely, we prove that there exist two positive constants and such that for any , the -norm of β-order derivatives of solutions obeys the following regularizing rates: for some suitable choices of s and q. These estimates particularly imply the spatial analyticity and the temporal decay of global solutions.

Asymptotic profile of solutions to the two-dimensional dissipative quasi-geostrophic equation

This paper is concerned with the asymptotic behavior of the two-dimensional dissipative quasi-geostrophic equation. Based on the spectral decomposition of the Laplacian operator and iterative techniques, we obtain improved L2 decay rates of weak solutions and derive more explicit upper bounds of higher order derivatives of solutions. We also prove the asymptotic stability of the subcritical quasi-geostrophic equation under large initial and external perturbations.

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 ℝ3. Based on linear Lp-Lq 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.

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 R 2 . Upper and lower bounds are derived for the L 2 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.

On analyticity rate estimates of the solutions to the Navier–Stokes equations in Bessel-potential spaces

On this paper spatial analyticity of solutions to the nonstationary incompressive Navier–Stokes flow in H ˙ n / 2 − 1 ( R n ) is established. The proof is based on the estimates for the higher order derivatives of solutions. These estimates imply not only the regularizing rates near t = 0 but also decaying rates at t → ∞ , as long as the solution exists. Although basic strategy is similar to our previous work with Giga for L n space, one can make the proof short using several tools from harmonic analysis.

On the smoothness of solutions of impulsive autonomous systems

The aim of this paper is to investigate dependence of solutions on parameters for nonlinear autonomous impulsive differential equations. We will specify what continuous, differentiable and analytic dependence of solutions on parameters is, define higher order derivatives of solutions with respect to parameters and determine conditions for existence of such derivatives. The theorem of analytic dependence of solutions on parameters is proved.

On the smoothness of solutions of impulsive autonomous systems

The aim of this paper is to investigate dependence of solutions on parameters for nonlinear autonomous impulsive differential equations. We will specify what continuous, differentiable and analytic dependence of solutions on parameters is, define higher order derivatives of solutions with respect to parameters and determine conditions for existence of such derivatives. The theorem of analytic dependence of solutions on parameters is proved.

Effects of Varying Nonlinearity and Their Singular Perturbation Flavour

Autonomous differential equations y″+f(y,p)=0 whose nonlinearity varies with a parameter p are studied. As a prototype, one may think of y″+f0(y)+|y|p−1g(y)=0. We discuss periodic solutions with initial values taken from various domains and their different types of convergence as p→∞. Equations y″−f(y,p)=0, yf(y,p)>0 are similarly discussed, with the “period” of a solution replaced by its “maximal interval of existence.” The study shows a natural link to singularly perturbed problems. It turns out that the family of ODEs under consideration are essentially a family of singularly perturbed problems. Solutions may develop “kinks” and higher order derivatives of solutions possess “boundary layers,” namely sets of non-uniform convergence. Similarities and differences between this family and the more common singularly perturbed problems which abound in the literature emerge.

A criterion for existence ofL 1-norms for higher-order derivatives of solutions to a homogeneous parabolic equation

A criterion for existence ofL 1-norms for higher-order derivatives of solutions to a homogeneous parabolic equation

Remark on the Rate of Decay of Higher Order Derivatives for Solutions to the Navier–Stokes Equations in [formula omitted

We present a new derivation of upper bounds for the decay of higher order derivatives of solutions to the unforced Navier–Stokes equations in Rn. The method, based on so-called Gevrey estimates, also yields explicit bounds on the growth of the radius of analyticity of the solution in time. Moreover, under the assumption that the Navier–Stokes solution stays sufficiently close to a solution of the heat equation in the L2 norm—a result known to be true for a large class of initial data—lower bounds on the decay of higher order derivatives can be obtained.

A priori estimates of higher order derivatives of solutions to the FitzHugh-Nagumo equations

In this paper we establish L ∞-bounds for the derivatives of all orders of the solutions to the FitzHugh-Nagumo equations, by means of comparison functions. We obtain bounds for the initial value problem, the Dirichlet problem and the Neumann problem. The FitzHugh-Nagumo equations arise in mathematical biology as a model for the conduction of electrical impulses along a nerve axon.