Global existence and analyticity of solution for the generalized Hall-magnetohydrodynamics system

We investigate the global existence and analyticity of mild solution to the three-dimensional generalized Hall-magnetohydrodynamics (MHD) system in this work. We prove the global existence and analyticity of solutions in the corresponding critical spaces. The work extends global existence and analyticity of solutions to Hall-MHD system in Duan and MHD system in Wang and Ye and Zhao, to the generalized Hall-MHD system with 1/2 ≤ α,β ≤ 1.     

Analyticity estimates for the Navier–Stokes equations

We study spatial analyticity properties of solutions of the three-dimensional Navier–Stokes equations and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier–Stokes equations with data in Hr, r?1/2, and prove a stability result for the analyticity radius.     

Global version of the Cauchy-Kovalevskaia theorem for nonlinear PDEs

The existence of global generalized solutions is proved for arbitrary analytic nonlinear PDEs on the whole of their domains of analyticity. The solutions are analytic outside of closed, nowhere dense subsets.     

On the Radius of Spatial Analyticity for the Inviscid Boussinesq Equations

In this paper, we study the problem of analyticity of smooth solutions of the inviscid Boussinesq equations. If the initial datum is real-analytic, the solution remains real-analytic on the existence interval. By an inductive method we can obtain a lower bound on the radius of spatial analyticity of the smooth solution.     

On the uniqueness and analyticity of solutions in micropolar thermoviscoelasticity

This paper deals with the linear theory of isotropic micropolar thermoviscoelastic materials. When the dissipation is positive definite, we present two uniqueness theorems. The first one requires the extra assumption that some coupling terms vanish; in this case, the instability of solutions is also proved. When the internal energy and the dissipation are both positive definite, we prove the well-posedness of the problem and the analyticity of the solutions. Exponential decay and impossibility of localization are corollaries of the analyticity.     

Existence and uniqueness for the Prandtl equations

Under the hypothesis of analyticity of the data with respect to the tangential variable we prove the existence and uniqueness of the mild solution of Prandtl boundary layer equation. This can be considered an improvement of the results of [8] as we do not require analyticity with respect to the normal variable.     

Remarks on analytic smoothing effect for the Schrödinger equation

We study analytic smoothing effect of solutions to the Schrödinger equation with Cauchy data decaying exponentially at infinity. The domain of analyticity in the space variables of solutions is described under weight conditions on the data in terms of the corresponding supporting function. The domain of analyticity in the time variable is characterized by means of weight conditions of Gaussian type on the data. A generalization of various isometrical identities related to the analytic smoothing effect is introduced.     

Stabilization in elastic solids with voids

In this paper we study the asymptotic behavior and the analyticity of the solutions of the one-dimensional porous-elasticity problem with thermal effect. Our main result is to prove the lack of exponential stability in case of the porous-elasticity with thermal effect when viscoelasticity is present. We prove the analyticity of the problem when a porous viscosity is present. We conclude by showing the impossibility of localization in time of the solutions in the isothermal case.     

Global solutions of Yang–Mills–Higgs systems on four-dimensional Minkowski space: II. Some properties of solutions

The paper concerns the properties of global solutions to the Cauchy problem of Yang–Mills–Higgs systems in the temporal gauge. The following properties are considered: global analyticity of solutions, an approximation property, conservation of the topological class of solutions, scattering of a part of the energy of solutions through a light cone.     

On regularizing-rate estimates for solutions to the heat flow with initial value problem

For a compact Riemannian manifold NRK without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from Rn to N, and the regularizing rate estimate of the strong solutions. Moreover, we obtain the analyticity in spatial variables of the solutions. The uniqueness of the mild solutions in C([0,T]; W1,n) is also considered in this paper.     

磁流体方程的时间解析性和时间周期解

考查了周期边界条件下的磁流体方程,证明了它的解关于时间是解析的,由此得到了磁流体方程的解的向后惟一性.对于周期解,证明了当周期小于某个常数时,周期的弱解是强解,进一步地这样的强解是定常解.     

(x)方程解的关于参数依赖性

本文作者研究拟凸域上的(x)-方程解关于参数的解析依赖性.     

The analyticity of solutions to a class of degenerate elliptic equations

In the present paper, the analyticity of solutions to a class of degenerate elliptic equations is obtained. A kind of weighted norms are introduced and under such norms some degenerate elliptic operators are of weak coerciveness.     

Remarks on shear flows of the Euler equations

The gradient estimates for renormalization solutions to the Euler equations are derived. The proof is based on the boundedness of the solutions to the linear transport equation, component-wisely. A shear flow is a unique globally-in-time strong solution in certain class due to the argument of renormalization solutions. Shear flows give lower bounds for gradient estimates as well as the analyticity rate. The threshold between locally well-posedness and ill-posedness of the Euler equations is clarified in terms of functions spaces of initial data.     

Radius of analyticity and exponential convergence for spectral projections of the generalized KdV equation

In this paper an exponential convergence rate for a spectral projection of the periodic initial-value problem for the generalized KdV equation is proved. Based on this convergence result, a method for determining the radius of analyticity of solutions of the generalized KdV equation is derived. Results from the new method and a similar method are compared.     

Quasi-analytic solutions of linear parabolic equations

Uniqueness of solutions of the Cauchy problem of a parabolic equation, and the related question of analyticity with respect to time, depend on global properties of the solution. We demonstrate that if the growth of the initial function (and, if relevant, of the inhomogeneous part and its derivatives) is not too great, solutions of non-stationary parabolic equations whose coefficients belong to quasi-analytic classes are quasi-analytic with respect to all variables and hence are unique. This study is motivated by the problem of endogenous completeness in continuous-time financial markets.     

Analyticity for Kuramoto–Sivashinsky‐type equations in two spatial dimensions

I. Stratis In this work, we investigate the analyticity properties of solutions of Kuramoto–Sivashinsky‐type equations in two spatial dimensions, with periodic initial data. In order to do this, we explore the applicability in three‐dimensional models of a spectral method, which was developed by the authors for the one‐dimensional Kuramoto–Sivashinsky equation. We introduce a criterion, which provides a sufficient condition for analyticity of a periodic function u∈C^∞, involving the rate of growth of ?ⁿu, in suitable norms, as n tends to infinity. This criterion allows us to establish spatial analyticity for the solutions of a variety of systems, including Topper–Kawahara, Frenkel–Indireshkumar, and Coward–Hall equations and their dispersively modified versions, once we assume that these systems possess global attractors. Copyright © 2015 John Wiley & Sons, Ltd.     

On the analyticity of solutions to semilinear differential equations degenerated on a submanifold

We investigate the analyticity of solutions to semilinear elliptic equations degenerated on a submanifold. We introduce a new weighted Sobolev space which is appropriate for studying such equations. The technique for linear equations using cut-off functions cannot be applied and we need to use a representation formula which requires a fundamental solution.     

Analyticity of Mild Solution for the 3D Incompressible Magneto-hydrodynamics Equations in Critical Spaces

The Cauchy problem for the 3D incompressible magneto-hydrodynamics equations in critcal spaces is considered. We first prove the global well-posedness of mild solution for the system in some time dependent spaces. Furthermore, we obtain analyticity of the solution.     

Analyticity and instabilities for interfaces: From Kelvin-Helmholtz instability to water waves

This paper is devoted to the analysis of solutions of fluid equations with interfaces and the purpose is to show how recent results on analyticity are related to the instabilities of the interface. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 15, Differential and Functional Differential Equations. Part 1, 2006.