Vector potential theory on nonsmooth domains in R<Superscript>3</Superscript> and applications to electromagnetic scattering

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space. The main goal is to develop the corresponding theory for L^p-integrable bounday data for optimal values of p's. We also discuss a number of relevant applications in electromagnetic scattering.     

Sur les équations différentielles ordinaires et les équations de transport

We study in this Note ordinary differential equations for divergence-free vector-fields with a limited regularity. We first observe that it is equivalent to solve the associated transport equations (i.e. Liouville equations). Then, we show existence, uniqueness, and stability results for generic vector-fields in L¹ or for “piecewise” W^1.1 vector-fields.     

Regularity theory for fully nonlinear integro‐differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C^{1, α} regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.     

Fusion transformations in Liouville theory

We study the fusion kernel for nondegenerate conformal blocks in the Liouville theory as a solution of difference equations originating from the pentagon identity. We propose an approach for solving these equations based on a “nonperturbative” series expansion that allows calculating the fusion kernel iteratively. We also find exact solutions for the special central charge values c = 1+6(b ? b^?1)², b ∈ ?. For c = 1, the obtained result reproduces the formula previously obtained from analytic properties of a solution of a Painlev´e equation, but our solution has a significantly simplified form.     

An LDLT factorization based ADI algorithm for solving large-scale differential matrix equations

Large-scale differential matrix equations appear in many applications like optimal control of partial differential equations, balanced truncation model order reduction of linear time varying systems etc. Here, we will focus on matrix Riccati differential equations (RDE). Solving such matrix valued ordinary differential equations (ODE) is a highly storage and time consuming process. Therefore, it is necessary to develop efficient solution strategies minimizing both. We present an LDL^T factorization based ADI method for solving algebraic Lyapunov equations (ALE) arising in the innermost iteration during the application of Rosenbrock ODE solvers to RDEs. We show that the LDL^T-type decomposition avoids complex arithmetic, as well as cancellation effects arising from indefinite right hand sides of the ALEs appearing in the classic ZZ^T based approach. Additionally, a certain number of linear system solves can be saved within the ADI algorithm by reducing the number of column blocks in the right hand sides while the full accuracy of the standard low-rank ADI is preserved. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Attractors and approximations for lattice dynamical systems

We present a sufficient condition for the existence of a global attractor for general lattice dynamical systems, then consider the existence of attractors and their approximation for second-order and first-order lattice systems which, in particular case, can be regarded as the spatial discretizations of corresponding wave equations and reaction-diffusion equations in R^k.     

Approximation of the incompressible non‐Newtonian fluid equations by the artificial compressibility method

This paper studies the approximation of the non‐Newtonian fluid equations by the artificial compressibility method. We first introduce a family of perturbed compressible non‐Newtonian fluid equations (depending on a positive parameter ε) that approximates the incompressible equations as ε → 0⁺. Then, we prove the unique existence and convergence of solutions for the compressible equations to the solutions of the incompressible equations. Copyright © 2012 John Wiley & Sons, Ltd.     

Local existence and blow‐up criterion for the generalized Boussinesq equations in Besov spaces

In this paper, we consider the three‐dimensional generalized Boussinesq equations, a system of equations resulting from replacing the Laplacian ? Δ in the usual Boussinesq equations by a fractional Laplacian ( ? Δ)^α. We prove the local existence in time and obtain a regularity criterion of solution for the generalized Boussinesq equations by means of the Littlewood–Paley theory and Bony's paradifferential calculus. The results in this paper can be regarded as an extension to the Serrin‐type criteria for Navier–Stokes equations and magnetohydrodynamics equations, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

Solution Blowup for Nonlinear Equations of the Khokhlov–Zabolotskaya Type

We consider several nonlinear evolution equations sharing a nonlinearity of the form ?²u²/?t². Such a nonlinearity is present in the Khokhlov–Zabolotskaya equation, in other equations in the theory of nonlinear waves in a fluid, and also in equations in the theory of electromagnetic waves and ion–sound waves in a plasma. We consider sufficient conditions for a blowup regime to arise and find initial functions for which a solution understood in the classical sense is totally absent, even locally in time, i.e., we study the problem of an instantaneous blowup of classical solutions.     

Anticipated backward stochastic differential equations with non-Lipschitz coefficients

This paper deals with a class of anticipated backward stochastic differential equations. We extend results of Peng and Yang (2009) to the case in which the generator satisfies non-Lipschitz condition. The existence and uniqueness of solutions for anticipated backward stochastic differential equations as well as a comparison theorem are obtained. The existence and uniqueness of L^p(p>2) solutions for anticipated backward stochastic differential equations are also studied.     

A note on the extended dispersionless Toda hierarchy

We derive dispersionless Hirota equations for the extended dispersionless Toda hierarchy. We show that the dispersionless Hirota equations are just a direct consequence of the genus-zero topological recurrence relation for the topological ?P¹ model. Using the dispersionless Hirota equations, we compute the twopoint functions and express the result in terms of Catalan numbers     

On the rate of convergence to equilibrium in the Becker-Döring equations

We provide an explicit rate of convergence to equilibrium for solutions of the Becker-Döring equations using the energy/energy-dissipation relation. The main difficulty is the structure of equilibria of the Becker-Döring equations, which do not correspond to a Gaussian measure, such that a logarithmic Sobolev-inequality is not available. We prove a weaker inequality which still implies for fast decaying data that the solution converges to equilibrium as e^−ct1/3.     

Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity

This paper is devoted to the investigation of the solution to the Cauchy problem for a system of partial differential equations describing thermoelasticity of nonsimple materials in a three-dimensional space. The model of linear dynamical thermoelasticity of nonsimple materials is considered as the system of partial differential equations of fourth order. In this paper, we proposed a convenient evolutionary method of approach to the system of equations of nonsimple thermoelasticity. We proved the L^p−L^q time decay estimates for the solution to the Cauchy problem for linear thermoelasticity of nonsimple materials.     

Pathwise random periodic solutions of stochastic differential equations

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify these as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general cases. We then use the argument of the relative compactness of Wiener-Sobolev spaces in C⁰([0,T],L²(Ω)) and generalized Schauder?s fixed point theorem to prove the existence of a solution of the coupled stochastic forward-backward infinite horizon integral equations. The condition on F is then further weakened by applying the coupling method of forward and backward Gronwall inequalities. The results are also valid for stationary solutions as a special case when the period τ can be an arbitrary number.     

CONSISTENCY OF THE FIELD EQUATIONS IN THE SCALE COVARIANT THEORY OF GRAVITATION

ABSTRACT

In the theory of general relativity, it is well known that the integrability conditions for Einstein's field equations are the conservation equations. The latter guarantee that if the constraint equations G^oa = kT^oa hold on an initial surface, and if G^av = kT^av at all times, then the constraint equations will hold at all earlier and later times. This ensures the consistency of the field equations and underlies the initial value approach to their solution. We investigate the corresponding result in the scale co-variant theory of gravitation.     

Geometric Schrdinger-Airy Flows on Khler Manifolds

We define a class of geometric flows on a complete Khler manifold to unify some physical and mechanical models such as the motion equations of vortex filament, complex-valued mKdV equations, derivative nonlinear Schrdinger equations etc. Furthermore, we consider the existence for these flows from S~1into a complete Khler manifold and prove some local and global existence results.     

Small perturbation solutions for nonlocal elliptic equations

We present a small perturbation result for nonlocal elliptic equations, which says that for a class of nonlocal operators, the solutions are in C^σ+α for any α∈(0,1) as long as the solutions are small. This is a nonlocal generalization of a celebrated result of Savin in the case of second order equations.     

Linear Equations and Regularity Conditions on Semigroups

R. Croisot in 1953, stated a very interesting problem of classification of all types of the regularity of semigroups defined by equations of the form a = a^mxaⁿ, with m,n \ge 0, m + n \ge 2. He proved that any of these equations determines either the ordinary regularity, left, right or complete regularity (see also the book by A. H. Clifford and G. B. Preston, Section 4.1). A similar problem, concerning all types of the regularity of semigroups and their elements defined by equations of the form a = a^pxa^qya^r, with p,q,r \ge 0, was treated by S. Lajos and G. Szász in 1975. The purpose of this paper is to generalize the results by Croisot, Lajos and Szász considering all types of the regularity of semigroups and their elements determined by more general equations called linear. We determine all types of the regularity of elements defined by linear equations, and prove that there are exactly 14 types of the regularity of semigroups defined by such equations. We also give implication diagrams for linear equations and regularity conditions.     

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.     

Semigroup Properties and the Crandall Liggett Approximation for a Class of Differential Equations with State-Dependent Delays

We present an approach for the resolution of a class of differential equations with state-dependent delays by the theory of strongly continuous nonlinear semigroups. We show that this class determines a strongly continuous semigroup in a closed subset of C^0, 1. We characterize the infinitesimal generator of this semigroup through its domain. Finally, an approximation of the Crandall-Liggett type for the semigroup is obtained in a dense subset of (C, ‖·‖_∞). As far as we know this approach is new in the context of state-dependent delay equations while it is classical in the case of constant delay differential equations.