Well-posedness analysis of a stationary Navier–Stokes hemivariational inequality

This paper provides a well-posedness analysis for a hemivariational inequality of the stationary Navier-Stokes equations by arguments of convex minimization and the Banach fixed point. The hemivariational inequality describes a stationary incompressible fluid flow subject to a nonslip boundary condition and a Clarke subdifferential relation between the total pressure and the normal component of the velocity. Auxiliary Stokes hemivariational inequalities that are useful in proving the solution existence and uniqueness of the Navier–Stokes hemivariational inequality are introduced and analyzed. This treatment naturally leads to a convergent iteration method for solving the Navier–Stokes hemivariational inequality through a sequence of Stokes hemivariational inequalities. Equivalent minimization principles are presented for the auxiliary Stokes hemivariational inequalities which will be useful in developing numerical algorithms.

On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term

The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation problem with a specified zero front in the case of two spatial variables. The solution existence and uniqueness theorem is proved in the class of analytic functions. The solution is constructed as a power series with coefficients to be calculated by a proposed constructive recurrent procedure. An algorithm based on the boundary element method using the dual reciprocity method is developed to solve the problem numerically. The efficiency of the application of the dual reciprocity method for various systems of radial basis functions is analyzed. An approach to constructing invariant solutions of the problem in the case of central symmetry is proposed. The constructed solutions are used to verify the developed numerical algorithm. The test calculations have shown the high efficiency of the algorithm.

Inverse problems for a class of linear Sobolev type equations with overdetermination on the kernel of operator at the derivative

The purpose of this work is to obtain sufficient conditions of a solution existence and uniqueness for a class of inverse problems for linear evolution equations with a degenerate operator at the derivative and with an unknown element in the right-hand side of the equation, which depends on the time variable. The overdetermination condition is given on the kernel of the operator at the derivative, the initial condition have the Cauchy form or the Showalter–Sidorov form. The obtained abstract results are applied to the investigation of linear inverse problems for the Sobolev system of equations and for the linearized Oskolkov system with overdetermination on the pressure gradient function.