Convergence to steady state of solutions of the euler equations,I

In this paper we analyze the convergence to steady state of solutions of the compressible and the incompressible isentropic Euler equations in two space dimensions. In the compressible case, the original equations do not converge. We replace the equation of continuity with an elliptic equation for the density, obtaining a new set of equations, which have the same steady solution. In the incompressible case, the equation of continuity is replaced by a Poisson equation for the pressure. In both cases, we linearize the equations around a steady solution and show that the unsteady solution of the linearized equations converges to the steady solution, if the steady solution is sufficiently smooth. In the proof we consider how the energy of the time dependent part developes with time, and find that it decrease exponentially.     

STATIONARY SOLUTION FOR A STOCHASTIC LINARD EQUATION WITH MARKOVIAN SWITCHING

This paper considers a stochastic Lienard equation with Markovian switching. The Feller continuity of its solution is proved by the coupling method and a truncation argument. The existence of a stationary solution for the equation is also proved under the Foster-Lyapunov drift condition.     

STATIONARY SOLUTION FOR A STOCHASTIC LI(E)NARD EQUATION WITH MARKOVIAN SWITCHING

This paper considers a stochastic Lienard equation with Markovian switching.The Feller continuity of its solution is proved by the coupling method and a truncation argument. The existence of a stationary solution for the equation is also proved under the Foster-Lyapunov drift condition.     

STATIONARY SOLUTION FOR A STOCHASTIC LIENARD EQUATION WITH MARKOVIAN SWITCHING

This paper considers a stochastic Liénard equation with Markovian switching. The Feller continuity of its solution is proved by the coupling method and a truncation argument. The existence of a stationary solution for the equation is also proved under the Foster-Lyapunov drift condition.     

Uniform operator continuity of the stationary Riccati equation in Hilbert space

In this paper the continuity in the uniform operator topology of the solution of the stationary Riccati equation in Hilbert space as a function of parameters is verified. The assumptions for this verification are the uniform operator continuity of the uncontrolled semigroup with respect to parameters, the uniform finiteness of the infimum of the quadratic cost functionals over the admissible controls, and uniform detectability. Some families of semigroups are described that satisfy the condition of continuity in the uniform operator topology with respect to parameters. The uniform operator continuity of the solution of the stationary Riccati equation with respect to parameters is important for applications to problems in adaptive control of stochastic evolution systems.This research was partially supported by NSF Grant ECS-8718026.     

Modulus of continuity of the solution to the Dirichlet problem for a second-order parabolic equation

The modulus of continuity of the solution to the Dirichlet problem is investigated for a second-order parabolic equation at a regular boundary point. A bound for the modulus of continuity is obtained in terms of the capacity. The coefficients of the equation are required to satisfy a Dini condition (uniformly).Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 587–593, April, 1976.     

On approximate solution of stochastic delay evolution equation in infinite dimensions

In [6] [7], the coefficients of given equation are supposed to be Lipschitz continuity or to be a bit weaker than Lipschitz continuity for establishing existence and uniqueness of the solution. In this paper, we show that, under the assumption of so-called pathwise uniqueness to the equation, a weaker restriction to coefficients than these mentioned above will recapture the convergence of Carathéodory approximation of the equation in the strong sense.     

Continuity properties of Hilbert space valued martingales

Necessary and sufficient conditions for Hölder continuity of Hilbert space valued martingales are given in terms of the associated quadratic variation. As an application one obtains a sufficient condition for a mild solution of a stochastic evolution equation to have a continuous version if the semigroup governing this equation is analytic. Further we derive Levy's modulus of continuity for the Hilbert space valued stochastic integral with the Wiener process as integrator and obtain a generalization of the loglog law for that integral.     

Construction of global‐in‐time solutions to Kolmogorov‐Feller pseudodifferential equations with a small parameter using characteristics

Using an idea going back to Madelung, we construct global in time solutions to the transport equation corresponding to the asymptotic solution of the Kolmogorov‐Feller equation describing a system with diffusion, potential and jump terms. To do that we use the construction of a generalized delta‐shock solution of the continuity equation for a discontinuous velocity field. We also discuss corresponding problem of asymptotic solution construction (Maslov tunnel asymptotics).     

On a stochastic fractional partial differential equation with a fractional noise

We will prove the existence, uniqueness and regularity of the solution for a stochastic fractional partial differential equation driven by an additive fractional space–time white noise. Moreover, the absolute continuity of the solution is also obtained.     

Quantum Cable Equations in Terms of Generalized Operators

A quantum cable equation in the sense of generalized operators is introduced. The existence and uniqueness of solutions are established and the continuity and a Markov-like property of the solution are obtained.     

External circular crack under arbitrary shear loading

An exact closed form solution in terms of elementary functions has been obtained to the governing integral equation of an external circular crack in a transversely isotropic elastic body. The crack is subjected to arbitrary tangential loading applied antisymmetrically to its faces. The recently discovered method of continuity solutions was used here. The solution to the governing integral equation gives the direct relationship between the tangential displacements of the crack faces and the applied loading. Now a complete solution to the problem, with formulae for the field of all stresses and displacements, is possible.     

Lipschitz continuity and semiconcavity properties of the value function of a stochastic control problem

We investigate the Cauchy problem for a nonlinear parabolic partial differential equation of Hamilton–Jacobi–Bellman type and prove some regularity results, such as Lipschitz continuity and semiconcavity, for its unique viscosity solution. Our method is based on the possibility of representing such a solution as the value function of the associated stochastic optimal control problem. The main feature of our result is the fact that the solution is shown to be jointly regular in space and time without any strong ellipticity assumption on the Hamilton–Jacobi–Bellman equation.     

On the modulus of continuity of the solution to the Dirichlet problem at a regular boundary point

A linear elliptic equation of second order with coefficients satisfying a Dini condition is considered in the paper. The modulus of continuity of a solution at a regular boundary point is investigated. An estimate for the modulus of continuity in terms of the Wiener capacity is obtained.Translated from Matematicheskie Zametki, Vol. 12, No. 1, pp. 67–72, July, 1972.     

Exponential self-similar mixing and loss of regularity for continuity equations

We consider the mixing behavior of the solutions to the continuity equation associated with a divergence-free velocity field. In this Note, we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity fields, thus showing the optimality of known lower bounds. We also describe how to use such examples to construct solutions to the continuity equation with Sobolev but non-Lipschitz velocity field exhibiting instantaneous loss of any fractional Sobolev regularity.     

四元数分析中T_Gf算子的Hlder连续性和Riemann-Hilbert边值问题

本文证明了四元数分析中的有界区域G上的非齐次Dirac方程u=f的分布解T_Gf,当f∈L_P(G),P>4时,在G上具有Holder连续性,讨论了超球和双圆柱上的方程u=f的Riemann-Hilbert边值问题,给出了可解条件和通解的积分表示,并且还证明了通解的Holder连续性。     

Lipschitz Continuity of Quasiconformal Mappings and of the Solutions to Second Order Elliptic PDE with Respect to the Distance Ratio Metric

The main aim of this paper is to study the Lipschitz continuity of certain \((K, K^{\prime })\)-quasiconformal mappings with respect to the distance ratio metric, and the Lipschitz continuity of the solution of a quasilinear differential equation with respect to the distance ratio metric.     

Hölder continuity of the solutions for a class of nonlinear SPDE’s arising from one dimensional superprocesses

The Hölder continuity of the solution X _t (x) to a nonlinear stochastic partial differential equation (see (1.2) below) arising from one dimensional superprocesses is obtained. It is proved that the Hölder exponent in time variable is arbitrarily close to 1/4, improving the result of 1/10 in Li et al. (to appear on Probab. Theory Relat. Fields.). The method is to use the Malliavin calculus. The Hölder continuity in spatial variable x of exponent 1/2 is also obtained by using this new approach. This Hölder continuity result is sharp since the corresponding linear heat equation has the same Hölder continuity.     

四元数分析中T_Gf算子的Hlder连续性和Riemann-Hilbert边值问题

本文证明了四元数分析中的有界区域G上的非齐次Dirac方程u=f的分布解T_Gf,当f∈L_P(G),P>4时,在G上具有Holder连续性,讨论了超球和双圆柱上的方程u=f的Riemann-Hilbert边值问题,给出了可解条件和通解的积分表示,并且还证明了通解的Holder连续性。     

Asymptotic analysis of singularly perturbed Hamilton-Jacobi equations

Unlike the previous investigation of the sufficient conditions for the convergence of minimax solutions of singularly perturbed Hamilton-Jacobi (H-J) equations, a typical example of which would be the Bellman-Isaacs (B-I) equations, convergence conditions are formulated not in terms of auxiliary constructs [1], but in terms of the Hamiltonian, the boundary function, assumptions regarding their continuity, Lipschitz continuity, etc. In addition, an asymptotic equation is derived, that is, a H-J equation whose minimax solution is the limit of solutions of H-J equations in which some of the momentum variables have coefficients whose denominators contain a small parameter which is made to approach zero.