Smooth periodic solutions of the penalized Maxwell fluid dynamics equations

The existence of smooth periodic solutions of the penalized Maxwell equations (2) is proved. The limit passage as ε→0 is investigated. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 206, 1993, pp. 85–90. Translated by Ya. Belopol'skaya.     

Galerkin method for linear parabolic equations with discontinuous solutions

Finite-element schemes are developed for solving a linear equation of parabolic type with a discontinuous solution in the space variable. Existence and uniqueness of the solution of the corresponding Cauchy problem is proved. Accuracy bounds are obtained for the finite-element scheme and the Crank-Nicholson scheme.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 33–40, 1988.     

Smoothness of solutions of second-order hyperbolic equations with discontinuous coefficients

Translated from Matematicheskie Zametki, Vol. 49, No. 6, pp. 3–11, June, 1991.     

On periodic solutions of ordinary differential equations with discontinuous right-hand side

A new version of the method of translation along trajectories, which does not require the uniqueness of the solution of the Cauchy problem, is applied to the proof of the existence theorem for vector-valued periodic solutions of ordinary differential equations of first and second order. This result is applicable to equations and differential inclusions with discontinuous right-hand side. Several applications of the theorems proved in this paper are considered in cases which are not covered by the classical theory of ordinary differential equations with continuous right-hand side and equations with right-hand side satisfying the Carathéodory conditions.     

Periodic solutions for second order differential equations with discontinuous restoring forces

We deal with the existence of periodic solutions for problems with a jump discontinuity. We use an approximation procedure and the method of the lower and upper solutions.     

Singularities of weak solutions of elliptic equations with discontinuous leading coefficients.

For the weak solutions of second-order equations in a plane bounded domain, one obtains the asymptotics and estimates of the solutions near the interior angular points of the line of discontinuity of the coefficients. It is established that the number of terms of the asymptotics, not in the Sobolev classW ₂ ² , can be arbitrarily large.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 47, pp. 166–169, 1974.     

On existence of solutions of multivalued stochastic differential equations with discontinuous coefficients

The subject of the paper is to find existence conditions of weak solutions to multivalued stochastic differential equations with discontinuous coefficients. First we prove that a non-exploding solution exists when the drift coefficient b satisfies linear growth and the diffusion coefficient σ is uniformly elliptic. On this basis, we continue to obtain a solution (up to the explosion time) in the weak sense under certain local integrability, improving the result of Rozkosz and S?omiński.     

Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients

We investigate linear and quasilinear evolutionary partial integro-differential equations of second order which include time fractional evolution equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems. We also show that a maximum principle holds for such equations.