Initial-boundary value problems for the Vlasov-Poisson equations in a half-space

We consider initial-boundary value problems for the Vlasov-Poisson equations in a half-space that describe evolution of densities for ions and electrons in a rarefied plasma. For sufficiently small initial densities with compact supports and large strength of an external magnetic field, we prove the existence and uniqueness of classical solutions for initial-boundary value problems with different boundary conditions for the electric potential: the Dirichlet conditions, the Neumann conditions, and nonlocal conditions.     

Initial-boundary value problems for incomplete singular perturbations of hyperbolic systems

The general problem studied has as a prototype the full non-linear Navier-Stokes equations for a slightly viscous compressible fluid including the heat transfer. The boundaries are of inflow-outflow type, i.e. non-characteristic, and the boundary conditions are the most general ones with any order of derivatives. It is assumed that the uniform Lopatinsky condition is satisfied. The goal is to prove uniform existence and boundedness of solution as the viscosity tends to zero and to justify the boundary layer asymptotics. The paper consists of two parts. In Part I the linear problem is studied. Here, uniform lower and higher order tangential estimates are derived and the existence of a solution is proved. The higher order estimates depend on the smoothness of coefficients; however this smoothness does not exceed the smoothness of the solution. In Part II the quasilinear problem is studied. It is assumed that for zero viscosity the overall initial-boundary value problem has a smooth solutionu ₀ in a time interval 0≦t≦T ₀. As a result the boundary laye, is weak and is uniformlyC ¹ bounded. This makes the linear theory applicable. an iteration scheme is set and proved to converge to the viscous solution. The convergence takes place for small viscosity and over the original time interval 0≦t≦T ₀.     

Some problems of scalar and vector-valued optimization in linear viscoelasticity

Optimal and Pareto-optimal stress rate histories are given for some simple scalar and vector-valued optimization problems in one-dimensional linear viscoelasticity.These problems were suggested to me by Néstor Distéfano only a few days before his untimely death. Support by ONR is gratefully acknowledged.     

Initial-boundary value problems for non-linear equations of generalized thermoelasticity and elasticity

We formulate a local existence theorem for the initial-boundary value problems of generalized thermoelasticity and classical elasticity. We present a unified approach to such boundary conditions as, for example, the boundary condition of traction, pressure or place combined with the boundary condition of heat flux or temperature.     

Initial-boundary value problems for anisotropic Landau-Lifshitz equations in three or four space dimensions

In this paper, we will show the existence of partially regular solutions to the initial-boundary value problem for Landau-Lifshitz equations with nonpositive anisotropy constants in three or four space dimensions. The partial regularity is proved up to the boundary both for the Dirichlet problem and for the Neumann problem. In addition, for the Neumann case, a generalized stability condition which ensures the partial regularity is given. For equations with positive or negative anisotropy coefficients, we will give two results of existence and uniqueness for the solutions corresponding to ground states.     

Growth estimates for solutions to initial-boundary value problems in viscoelasticity

For the abstract Volterra integro-differential equation u_tt ? Nu + ∝_?∞^t K(t ? τ) u(τ) dτ = 0 in Hilbert space, with prescribed past history u(τ) = U(τ), ? ∞ < τ < 0, and associated initial data u(0) = f, u_t(0) = g, we establish conditions on K(t), ? ∞ < t < + ∞ which yield various growth estimates for solutions u(t), belonging to a certain uniformly bounded class, as well as lower bounds for the rate of decay of solutions. Our results are interpreted in terms of solutions to a class of initial-boundary value problems in isothermal linear viscoelasticity.     

Boundary value problems of the theory of viscoelasticity for Kelvin-Voight materials

In this paper, the classical Kelvin-Voight model of the linear theory of viscoelasticity is considered and the following results are obtained: the fundamental solution of the equation of steady vibrations is constructed, the basic properties of plane waves and elastopotentials are established, the uniqueness theorem of the internal and external boundary value problems (BVPs) are proved, the existence theorems for classical solutions of the BVPs by means of the potential method and the theory of two-dimensional singular integral equations are proved. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Initial-boundary value problems for a class of pseudoparabolic equations with integral boundary conditions

In this paper, we deal with a class of pseudoparabolic problems with integral boundary conditions. We will first establish an a priori estimate. Then, we prove the existence, uniqueness and continuous dependence of the solution upon the data. Finally, some extensions of the problem are given.     

Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems

We consider the usual linear elastodynamics equations augmented with evolution equations for viscoelastic internal stresses. A fully discrete approximation is defined, based on a spatially symmetric or non‐symmetric interior penalty discontinuous Galerkin finite element method, and a displacement‐velocity centred difference time discretisation. An a priori error estimate is given but only the main ideas in the proof of the error estimate are reported here due to the large number of (mostly technical) estimates that are required. The full details are referenced to a technical report. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On a method of solving two-dimensional problems of the linear theory of viscoelasticity

We describe the basic propositions of the linear theory of viscoelasticity. We give transformation formulas for the resolvent integral operators of viscoelasticity with an arbitrary analytic kernel of difference type. The method of computing the irrational operator functions is illustrated by determining the real parameters of the two-dimensional stressed state of an orthotropic plate. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 86–96.     

Use of the results of auxiliary experiments in solving problems in the linear theory of viscoelasticity

The approximate method of solving problems of the theory of linear viscoelasticity with arbitrary creep and relaxation kernels, proposed in [2], is substantiated and generalized. The essence of this method consists in the approximation of the functions depending on the Laplace — Carson transforms of the mechanical characteristics of a viscoelastic body by means of certain combinations of the transforms of the creep and relaxation kernels. The expressions obtained as a result of the approximation enable the inverse transforms of the unknown functions to be found without difficulty.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 963–969, November–December, 1968.     

Stress obtained by interpolation methods for a boundary value problem in linear viscoelasticity

We will solve the following boundary value problem in linear viscoelasticity: given the value of the stress on (a part of) the boundary of the domain find the stress in the whole body at all positive times. Using a state space setting we show that the stress field inside the body can be obtained from the boundary stress by a variation-of-parameters formula involving an analytic semigroup. The relation between the regularities of the boundary stress and the stress inside the body is therefore characterized by the well-known and rich regularity theory for analytic semigroups.     

Initial-boundary value problem for the nonlinear string-beam system

This paper deals with a nonlinear string-beam system describing the torsional-vertical oscillations of a suspension bridge. We consider the initial-boundary value problem and study the existence and uniqueness question. We assume time independent right hand sides, but allow quite general nonlinear terms. Using the Faedo-Galerkin method we prove the existence of a unique solution on an arbitrary large time interval.     

Asymptotically linear elliptic boundary value problems

We study semilinear problems in which the nonlinear term has different asymptotic behavior at ± with the limits (1.2) spanning a finite number of eigenvalues of the linear operator.Research supported in part by an NSF grant.