Relations between regularity coefficients

For a bounded sequence of matrices defining a nonautonomous dynamics with discrete time, we obtain all possible relations between the regularity coefficients introduced by Lyapunov, Perron and Grobman. This includes considering general inequalities between the coefficients and showing that these inequalities are the best possible, in the sense that for any three nonnegative numbers satisfying them, and for no others, there exists a bounded sequence of matrices having the numbers respectively as Lyapunov, Perron and Grobman coefficients. Moreover, we establish inequalities between the three coefficients and some other regularity coefficients.     

On the regularity of discrete linear systems

In this paper we have introduced a new regularity coefficient of time varying discrete linear system. On the base of this coefficient we have characterized the regularity of homogeneous discrete time varying linear systems by nonhomogeneous ones. Moreover we provided bounds for the regularity coefficient in terms of the existing in the literature regularity coefficients.     

Regularity of functions smooth along foliations, and elliptic regularity

We produce two sets of results arising in the analysis of the degree of smoothness of a function that is known to be smooth along the leaves of one or more foliations. These foliations might arise from Anosov systems, and while each leaf is smooth, the leaves might vary in a nonsmooth fashion. One set of results gives microlocal regularity of such a function away from the conormal bundle of a foliation. The other set of results gives local regularity of solutions to a class of elliptic systems with fairly rough coefficients. Such a regularity theory is motivated by one attack on the foliation regularity problem.     

On the C^1 Regularity of Solutions to Divergence Form Elliptic Systems with Dini-Continuous Coefficients

The author proves C1 regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients.     

Evolution systems for paraxial wave equations of Schrödinger-type with non-smooth coefficients

We prove existence of strongly continuous evolution systems in L² for Schrödinger-type equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with additional pseudodifferential terms in time and low regularity in the lateral space variables. We formulate and analyze the Cauchy problem in distribution spaces with mixed regularity. The key point in the evolution system construction is an elliptic regularity result, which enables us to precisely determine the common domain of the generators. The construction of a solution with low regularity in the coefficients is the basis for an inverse analysis which allows to infer the lack of lateral regularity in the medium from measured data.     

On the C~1 Regularity of Solutions to Divergence Form Elliptic Systems with Dini-Continuous Coefficients(Dedicated to Haim Brezis on his 70th birthday with friendship and admiration)

The author proves C~1 regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients.     

Symmetrisers and generalised solutions for strictly hyperbolic systems with singular coefficients

This paper is devoted to strictly hyperbolic systems and equations with non‐smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate Gårding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of the Cauchy problem (or of a transmission problem in the piecewise regular case) exists, the generalised solution is shown to be associated with the classical solution (or the piecewise classical solution satisfying the appropriate transmission conditions).     

Hölder continuity in time for SG hyperbolic systems

We investigate well posedness of the Cauchy problem for SG hyperbolic systems with non-smooth coefficients with respect to time. By assuming the coefficients to be Hölder continuous we show that this low regularity has a considerable influence on the behavior at infinity of the solution as well as on its regularity. This leads to well posedness in suitable Gelfand-Shilov classes of functions on Rn. A simple example shows the sharpness of our results.     

Some results in Floquet theory,with application to periodic epidemic models

In this paper, we present several new results to the classical Floquet theory on the study of differential equations with periodic coefficients. For linear periodic systems, the Floquet exponents can be directly calculated when the coefficient matrices are triangular. Meanwhile, the Floquet exponents are eigenvalues of the integral average of the coefficient matrices when they commute with their antiderivative matrices. For the stability analysis of constant and nontrivial periodic solutions of nonlinear differential equations, we derive a few results based on linearization. We also briefly discuss the properties of Floquet exponents for delay linear periodic systems. To demonstrate the application of these analytical results, we consider a new cholera epidemic model with phage dynamics and seasonality incorporated. We conduct mathematical analysis and numerical simulation to the model with several periodic parameters.     

具有Dini连续性系数的非线性椭圆方程组在自然增长条件下的最优内部部分正则性

本文我们研究的是具有Dini连续性系数的散度形式的非线性椭圆方程组在自然增长条件下的问题．我们证明所用的方法是有Dugaar和Grotowski所引进的调和逼近技巧。这种技窍在证明弱解的局部正则性时非常重要．我们可以用之直接得到最优局部正则性结果．     

Global wellposedness for quasi-linear symmetric hyperbolic systems with regularized coefficients

Following the abstract setting of [8] and using the global results of [2], global wellposedness and regularity results are proved for the solutions of quasi-linear symmetric hyperbolic systems with bounded coefficients which are regularized by a convolution in the space variables with a regularizing function. In the case of unbounded regularized coefficients, local existence of classical solutions is proved, as well as uniqueness and regularity of (not necessarily existing) global weak solutions with initial value in a Sobolev space. As the regularizing function tends to Dirac's δ, local-in-time convergence to the classical solution of the non-regularized problem is proved.     

Regularity of Solutions to a Diffraction-Type Problem for Nondiagonal Linear Elliptic Systems in the Campanato Space

We study a two-phase difraction-type problem for nondiagonal linear elliptic systems of equations. We study the regularity of a weak solution to the problem in the Campanato spaces. In particular, we prove the smoothness of solutions in a neighborhood of the interface surface separating media. For model systems with constant coefficients we derive Campanato-type integral estimates. Bibliography: 13 titles.     

Global Regularity of Solutions to Systems of Reaction–Diffusion with Sub-Quadratic Growth in Any Dimension

This paper is devoted to the study of the regularity of solutions to some systems of reaction–diffusion equations, with reaction terms having a subquadratic growth. We show the global boundedness and regularity of solutions, without smallness assumptions, in any dimension N. The proof is based on blow-up techniques. The natural entropy of the system plays a crucial role in the analysis. It allows us to use of De Giorgi type methods introduced for elliptic regularity with rough coefficients. In spite these systems are entropy supercritical, it is possible to control the hypothetical blow-ups, in the critical scaling, via a very weak norm. Analogies with the Navier–Stokes equation are briefly discussed in the introduction.     

Geometry of quasilinear hyperbolic systems with characteristic fields of constant multiplicity

In this paper, we study the regularity of the eigenvalues and eigenvectors and the existence of normalized coordinates for quasilinear hyperbolic systems with characteristic fields of constant multiplicity. We prove that the eigenvalues and eigenvectors of the system have the same regularity as the coefficients of the system. On the other hand, we show that, for the quasilinear hyperbolic system of conservation laws with characteristic fields of constant multiplicity, the normalized coordinates exist on the domain under consideration.     

On Some Properties of the Behavior of Linear Extensions of Dynamical Systems on a Torus under Perturbation of Phase Variables

We investigate classes of linear extensions of dynamical systems on a torus for which the Lyapunov functions exist for an arbitrary flow on the torus. Linear extensions for which the Lyapunov functions exist only with varying coefficients are considered separately. We investigate the problem of preservation of regularity under perturbation of phase variables.     

Hypoelliptic systems connected with Newton's polyhedron

We introduce a general class of hypoelliptic systems of partial differential operators with variable coefficients generalizing the so‐called multi‐quasi‐elliptic differential operators. We study the regularity of solutions of these systems. More complete results are obtained in the case of two independent variables. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gain of regularity for semilinear Schrödinger equations

We discuss local existence and gain of regularity for semilinear Schr?dinger equations which generally cause loss of derivatives. We prove our results by advanced energy estimates. More precisely, block diagonalization and Doi's transformation, together with symbol smoothing for pseudodifferential operators with nonsmooth coefficients, apply to systems of Schr?dinger-type equations. In particular, the sharp G?rding inequality for pseudodifferential operators whose coefficients are twice continuously differentiable, plays a crucial role in our proof. Received: 14 December 1998     

Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains

This note establishes regularity estimates for the solution of the Maxwell equations in Lipschitz domains with non-smooth coefficients and minimal regularity assumptions. The argumentation relies on elliptic regularity estimates for the Poisson problem with non-smooth coefficients.     

Orlicz regularity for higher order parabolic equations in divergence form with coefficients in weak BMO

We consider higher order parabolic equations in divergence form with measurable coefficients to find optimal regularity in Orlicz spaces of the maximum order derivatives of the weak solutions. The relevant minimal regularity requirement on the tensor matrix coefficients is of small BMO in the spatial variable and is measurable in the time variable. As a consequence we prove the classical W ^m,p regularity, m = 1, 2, . . . , 1 < p < ∞, for such higher order equations. In the same spirit the results easily extend to higher order parabolic systems as well as up to the boundary.