单边算子有界性和KdV型色散方程的适定性研究

数学和物理中许多重要问题均可归结为算子在某些函数空间中的有界性质.奇异积分算子有界性质的研究是调和分析理论的核心课题之一,由此发展起来的各种方法和技巧已广泛应用于偏微分方程的研究.借助奇异积分算子在Lebesgue空间或Morrey型空间中建立的时空估计和半群理论,可以得到非线性色散方程在低阶Sobolev空间中Cauchy问题的适定性.本文首次定义一类单边振荡奇异积分算子并研究该类算子的经典加权有界性质.受经典交换子刻画理论的启发,本文首次引入Morrey空间的交换子刻画理论.利用不同于常规极大函数的方法得到两类象征函数在Morrey空间中的交换子刻画.以上结果为偏微分方程的研究提供了新的工具.最后,结合能量方法和数论知识,本文解决几类KdV型色散方程的适定性问题.     

Abstract Elliptic Equations with Integral Boundary Conditons

This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in $L_{p}$ spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed $L_{\mathbf{p}}$ norm.     

Maximal regularity of type L p for abstract parabolic Volterra equations

We prove maximal regularity results of type L_p for abstract parabolic Volterra equations including problems with inhomogeneous boundary data. Our approach is purely operator theoretic. It uses the inversion of the convolution, the Dore-Venni theorem, the Mikhlin theorem in the operator-valued version, and real interpolation. Known results on L_p-regularity of abstract Cauchy problems and abstract parabolic pdes with inhomogeneous boundary conditions are recovered. As an application we consider the heat equation of memory type with inhomogeneous boundary condition.     

Phase space analysis of semilinear parabolic equations

We present a wave packet analysis of a class of possibly degenerate parabolic equations with variable coefficients. As a consequence, we prove local wellposedness of the corresponding Cauchy problem in spaces of low regularity, namely the modulation spaces, assuming a nonlinearity of analytic type. As another application, we deduce that the corresponding phase space flow decreases the global wave front set. We also consider the action on spaces of analytic functions, provided the coefficients are analytic themselves.     

Strichartz estimates for parabolic equations with higher order differential operators

The present paper first obtains Strichartz estimates for parabolic equations with nonnegative elliptic operators of order 2m by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-LittlewoodSobolev inequality. Some conclusions can be viewed as the improvements of the previously known ones. Furthermore, an endpoint homogeneous Strichartz estimates on BMOx(Rn) and a parabolic homogeneous Strichartz estimate are proved. Meanwhile, the Strichartz estimates to the Sobolev spaces and Besov spaces are generalized. Secondly, the local well-posedness and small global well-posedness of the Cauchy problem for the semilinear parabolic equations with elliptic operators of order 2m, which has a potential V(t, x) satisfying appropriate integrable conditions, are established. Finally, the local and global existence and uniqueness of regular solutions in spatial variables for the higher order elliptic Navier-Stokes system with initial data in Lr(Rn) is proved.     

Existence of Global Solutions for a Class of Abstract Second-Order Nonlocal Cauchy Problem with Impulsive Conditions in Banach Spaces

In this paper, we are concerned with a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. First, we study the existence of mild solutions for a class of second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces on an interval [0,a]. Later, we study a couple of cases where we can establish the existence of global solutions for a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. We apply our theory to study the existence of solutions for impulsive partial differential equations.     

Unique solvability of a non‐local problem for mixed‐type equation with fractional derivative

In this work, we investigate a boundary problem with non‐local conditions for mixed parabolic–hyperbolic‐type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.     

A class of degenerate parabolic equations with a concentrated nonlinear source

In this paper, we consider the Cauchy problem for a class of degenerate parabolic equations with a concentrated nonlinear source. We obtain the existence of the generalized solutions for the problem based on some a priori estimates on solutions.     

Integrated semigroups and parabolic equations. Part I: linear perburbation of almost sectorial operators

The paper deals with linear abstract Cauchy problem with non-densely defined and almost sectorial operators, whenever the part of this operator in the closure of its domain is sectorial. This kind of problem naturally arises for parabolic equations with non-homogeneous boundary conditions. Using the integrated semigroup theory, we prove an existence and uniqueness result for integrated solutions. Moreover, we study the linear perturbation problem.     

The Construction of an Evolution System in the Hyperbolic Case and Applications

We study the Cauchy problem for abstract linear and quasi–linear non–autonomous evolution equations of hyperbolic type using semigroup theory. Under weak differentiability assumptions on the time regularity of the coefficients we prove well–posedness and regularity of a solution. The abstract results are illustrated by their application to a series of equations of mathematical physics.     

A mollification method for ill-posed problems

Summary. A mollification method for a class of ill-posed problems is suggested. The idea of the method is very simple and natural: if the data are given inexactly then we try to find a sequence of ``mollification operators" which map the improper data into well-posedness classes of the problem (mollify the improper data). Within these mollified data our problem becomes well-posed. And when these facts are in hand we try to obtain error estimates and optimal or ``quasi-optimal" mollification parameters. The method is working not only for problems in Hilbert spaces, but also for problems in Banach spaces. Applications of the method to concrete problems, like numerical differentiation, parabolic equations backwards in time, the Cauchy problem for the Laplace equation, one- and multidimensional non-characteristic Cauchy problems for parabolic equations (in infinite or finite domains),... give us very sharp stability estimates of H\"older continuous type. In these cases the method is optimal in the sense that it gives the same order of H\"older continuous dependence on the data as for the regularized problems. Furthermore, the method may be implemented numerically using fast Fourier transforms. For the first time a uniform stability estimate of H\"older continuous type of the solution of the heat equation backwards in time in the space for all could be established by our mollification method. A new simple sharp pointwise estimate of H\"older type for the weak solution of a non-characteristic Cauchy problem for parabolic equations in a finite domain is established. Received June 25, 1993 / Revised version received February 18, 1994     

A continuous dependence result for ultraparabolic equations in option pricing

We prove continuous dependence results for solution to the Cauchy problem related to degenerate parabolic equations arising in the valuation of financial derivatives. These results are crucial in some standard calibration procedure for recent stochastic volatility and interest rates models.     

Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator

We study an abstract nonlinear evolution equation governed by a time-dependent operator of subdifferential type in a real Hilbert space. In this paper we discuss the convergence of solutions to our evolution equations. Also, we investigate the optimal control problems of nonlinear evolution equations. Moreover, we apply our abstract results to a quasilinear parabolic PDE with a mixed boundary condition.     

Cauchy problem for parabolic systems with convolution operators in periodic spaces

For a class of periodic systems of parabolic type with pseudodifferential operators containing $\{ \vec p,\vec h\} $ -parabolic systems of partial differential equations, we study the properties of the fundamental matrices of the solutions and establish the well-posed solvability of the Cauchy problem for these systems in the spaces of generalized periodic functions of the type of Gevrey ultradistributions. For a particular subclass of systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.     

The Fredholm alternative for parabolic evolution equations with inhomogeneous boundary conditions

We study the Fredholm properties of parabolic evolution equations on R with inhomogeneous boundary values. These problems are transformed into evolution equations with inhomogeneities taking values in certain extrapolation spaces. Assuming that the underlying homogeneous problem is asymptotically hyperbolic, we show the Fredholm alternative for these equations. The results are applied to parabolic partial differential equations.     

On asymptotics of solutions of parabolic equations with nonlocal high-order terms

In the paper, we study the Cauchy problem for second-order differential-difference parabolic equations containing translation operators acting to the high-order derivatives with respect to spatial variables. We construct the integral representation of the solution and investigate its long-term behavior. We prove theorems on asymptotic closeness of the constructed solution and the Cauchy problem solutions for classical parabolic equations; in particular, conditions of the stabilization of the solution are obtained. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 143–183, 2005.     

Nonsmooth analysis of doubly nonlinear evolution equations

In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional, for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.     

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study Plemelj-Calderón-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’ monogenics and show that this problem has finite index. We also consider Szegö projections and the corresponding L^p-decompositions. Our approach relies on an extension of the classical Calderón-Zygmund theory of singular integral operators which allow one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.     

Evolution equations for abstract differential operators

We study in this paper the wellposedness and regularity of solutions of evolution equations associated with abstract differential operators on a Banach space. The results can be applied to many partial differential equations on different function spaces.     

On nonlocal solutions of semilinear equations of the Sobolev type

We prove the existence and uniqueness of a classical solution of the Cauchy problem and the Showalter problem for some semilinear equations of the Sobolev type on a given interval. The abstract results are used to study initial-boundary value problems for some systems of equations unsolvable for the time derivative.