Boundary-value problem for a mixed parabolic-hyperbolic equation of the third order

The unique solvability of a certain boundary-value problem is proved for a mixed parabolic-hyperbolic type equation of the third order.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 20–29, January, 1995.     

Stokes’ first problem for a third grade fluid in a porous half space

In this study, we model the flow of a third grade fluid in a porous half space. Based on modified Darcy’s law, the flow over a suddenly moved flat plate is discussed numerically. The influence of various parameters of interest on the velocity profile is seen.     

First mixed problem for a nonstrictly hyperbolic equation of the third order in a bounded domain

We study the classical solution of a boundary value problem for a nonstrictly parabolic equation of the third order in a rectangular domain of two independent variables. We pose Cauchy conditions on the lower base of the domain and the Dirichlet conditions on the lateral boundary. By the method of characteristics, we obtain a closed-form analytic expression for the solution of the problem. The uniqueness of the solution is proved.     

The mixed problem for a degenerate second-order hyperbolic equation

We study conditions for uniqueness and existence of a solution for the mixed problem for a second-order hyperbolic equation that is degenerate at a finite instant of time.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 39–42.     

Existence for a characteristic boundary value problem for an equation of mixed type in three-dimensional space

The equation of mixed type With k(x₃) = sign x₃|x₃|^m, m > 0, d?C¹(?), x = (x₁, x₂, x₃), is considered in the threedimensional region G which is bounded by the surfaces: a piecewise smooth surface Γ₀ lying in the half-space x₃ > 0 which intersects the plane x₃ = 0 in the unit circle, and for x₃ < 0 by the characteristic surfaces We prove existence of a generalized solution for the characteristic boundary value problem: Lu = fin G, u_Γ0∪Γ1 = 0. The result is obtained by using a variant of the energy-integral method.     

Weakly singular initial values for the Stokes equation on a half space

We study the extension of Ukai's formula to the case of singular initial values for the Stokes problem on the half space.     

A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

First mixed problem for a parabolic difference-differential equation

The first mixed boundary value problem for a parabolic difference-differential equation with shifts with respect to the spatial variables is considered. The unique solvability of this problem and the smoothness of generalized solutions in some cylindrical subdomains are established. It is shown that the smoothness of generalized solutions can be violated on the interfaces of neighboring subdomains. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 145–153, July, 1999.     

$ L^p $-Theory of the Stokes equation in a half space

In this paper, we investigate -estimates for the solution of the Stokes equation in a half space H where . It is shown that the solution of the Stokes equation is governed by an analytic semigroup on or . From the operatortheoretical point of view it is a surprising fact that the corresponding result for does not hold true. In fact, there exists an -function f satisfying such that the solution of the corresponding resolvent equation with right hand side f does not belong to . Taking into account however a recent result of Kozono on the nonlinear Navier-Stokes equation, the -result is not surprising and even natural. We also show that the Stokes operator admits a R-bounded -calculus on for 1 < p < and obtain as a consequence maximal -regularity for the solution of the Stokes equation. Received August 24, 2000; accepted September 30, 2000.     

On a Frankl-type problem for a mixed parabolic-hyperbolic equation

We state a new nonlocal boundary value problem for a mixed parabolic-hyperbolic equation. The equation is of the first kind, i.e., the curve on which the equation changes type is not a characteristic. The nonlocal condition involves points in hyperbolic and parabolic parts of the domain. This problem is a generalization of the well-known Frankl-type problems. Unlike other close publications, the hyperbolic part of the domain agrees with a characteristic triangle. We prove unique solvability of this problem in the sense of classical and strong solutions.     

The mixed problem for laplace's equation in a class of lipschitz domains

In the focusing problem for the radially symmetric porous medium equation, one starts with initial data supported outside a ball centered at the origin, and studies the flow unitl the focusing thim, i.e., until the moment when the support of the solution reaches the origin. For any fixed focusing time, say t = 0, there exists a one-parameter family {g_c(r,t)} of self-similar solutions to the focusing problem. We prove that if V(r,t) is a radially symmetric porous medium pressure such that supp V(·,t₀) = [a,b]?R⁺ for some t₀<0 and V focuses at t = 0, then there exist a c^*?R⁺ such that (in the appropriate technical sense) V is approximated by g_c* for (r,t) near (0,0).