Nonlocal problems for the equations of motion of Kelvin-Voight fluids

For the equations of the motion of Kelvin-Voight fluids (0.1) and for the semilinear abstract differential Eqs. (0.11)–(0.17) arising in the theory of Sobolev equations (to which Eqs. (0.1) also belong), we study the four following nonlocal problems: 1) the solvability of the initial boundary problem for Eqs. (0.1) and the Cauchy problem for Eqs. (0.11)–(0.17) on the semiaxis 0<t≤∞; 2) the existence of periodic solutions of Eqs. (0.1) and Eqs. (0.11)–(0.17) with a free term periodic in t; 3) exponential stability theory for solutions of Eqs. (0.1) and Eqs. (0.11)–(0.17) as t→∞ and related problems; 4) attractor theory for Eqs. (0.1). Bibliography: 40 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 120–158, 1992.     

Nonlocal problems in the theory of the motion equations of Kelvin-Voight fluids

For the motion equations of Kelvin-Voight fluids one proves: 1) a global theorem for the existence and uniqueness of a solution (v;{u_e}) of the initial-boundary value problem on the semiaxis t R⁺ from the class W ¹ (R⁺); W ₂ ² () H()) with initial condition v_o(x) W ₂ ² () H() when the right-hand side f(x, t) L(R ⁺; L₂()); 2) a global theorem for the existence and uniqueness of a solution (v; {u_l}) on the entire axisR from the classW ¹ (R; W ₂ ² () H()) when the right-hand side f(x, t) L(R; L₂()); 3) a global theorem for the existence of at least one solution (v; {u_l}), periodic with respect to t with period , from the class W ¹ (R ⁺; W ₂ ² () H()) when the right-hand side f(x, t) L(R ⁺; L₂()) is periodic with respect to t with period , and a local uniqueness theorem for such a solution; 4) a theorem for the existence and uniqueness in the small of a solution (v; {u_l}), almost periodic with respect to t R, from V. V. Stepanov's class S ¹ (R; W ₂ ² ()H()) when the right-hand side f(x, t) S(R; L₂()) is almost periodic with respect to t; 5) the linearization principle (Lyapunov's first method) is justified in the theory of the exponential stability of the solutions of an initial-boundary value problem in the space H() and conditions are given for the exponential stability of a stationary and periodic solution, with respect to t R, of the system (1).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 181, pp. 146–185, 1990.     

Nonlocal problems in the theory of equations of motion of Kelvin-Voight fluids. 2

This article studies nonlocal problems for equations of motion of Kelvin-Voight fluids (2): 1) global solvability of initial-boundary-value problem (2)-(3) on halfaxisR ⁺ with free termf(x, t) S₂(⁺; L₂(0)) (see (4)); 2global solvability of system (2) on the entire axis R in the class of functions that are bounded as t±with free term f(x,t)S₂(; L₂());3) the existence of periodic solutions for system (2) that are periodic in t with period with free term,f(x,t)L₂((0,); L₂());4) the existence of solutions of system (2) that are almost periodic in t with free term f(x,t)S₂(, L₂()).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 111–124, 1990.     

Dynamical system generated by the equations of motion of Oldroyd fluids

An attractor m is constructed for the two-dimensional initial-boundary problem for the equations of motion of Oldroyd's fluid, the properties of the evolution operator V_t, t0, are investigated, and the dynamical system {m; V_t, –t} is constructed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 136–141, 1986.The authors express gratitude to L. D. Faddeev and O. A. Ladyzhenskava for supporting their research on the hydrodynamics of non-Newtonian fluids.     

Interaction of kinks for semilinear wave equations with a small parameter

We consider a class of semilinear wave equations with a small parameter and nonlinearities such that the equations have exact kink-type solutions. The main result consists in obtaining sufficient conditions for the nonlinearities under which the interaction of kinks preserves the sine-Gordon scenario. This means that the interaction occurs without changing the waves shape and with shifts of trajectories.     

Asymptotic stability and time periodicity of "small" solutions of the equations of the motion of oldroyd and Kelvin-Voight fluids

One proves the asymptotic stability and the time periodicity of the "small" classical solutions of the systems of equations (1) and (2), describing the motion of the Oldroyd and Kelvin—Voight fluids, respectively.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 63–75, 1990.     

Limit behavior and on the attractor for the equations of motion of Oldroyd fluids

One constructs the attractor m, of a two-dimensional initial-boundary-value problem for the equations of motion of Oldroyd fluids, one proves the finite-dimensionality of the dynamical problem on m and the finiteness of the Hausdorff dimension of the attractor m.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 67–71, 1986.The authors express their gratitude to O. A. Ladyzhenskaya for her interest in the paper and for useful discussions.     

On parabolic equations with a small parameter

A singularly perturbed parabolic equation with a nonlinear right-hand side of a special form is examined. A numerical analytical study of such equations is performed.     

Smooth convergent ε-approximations of the first initial boundary-value problem for the equations of motion of Kelvin-Voight fluids

Smooth convergent ε-approximations (33), (34) and (38), (39) are constructed for the equations of Kelvin-Voight fluids (1) and for the equations of Kelvin-Voight fluids of order L=1,2,…(2). Namely, it is shown that, for any ε>0, the first initial boundary-value problem for the three-dimensional perturbed systems (33), (34) and (38), (39) has a unique classical solution in the large. As ε→0, these solutions converge to the classical solutions of the first initial boundary-value problem for the equations of Kelvin-Voight fluids (1) and for the equations of Kelvin-Voight fluids of order L=1,2,… (2), respectively. Bibliography: 19 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 186–212. Translated by N. B. Lebedinskaya.     

Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations

The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter.

     

On a bounded almost periodic solution of a semilinear parabolic equation

We obtain sufficient conditions for the existence and uniqueness of a bounded almost periodic solution of a semilinear parabolic equation. Institute of Theoretical and Applied Mathematics, Ministry of Science and Academy of Sciences of Kazakhstan, Alma-Ata, Kazakhstan. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 828–830, June, 2000.