The topological degree method for equations of the Navier‐Stokes type

Abstract

Abstract. We obtain results of existence of weak solutions in the Hopf sense of the initial-boundary value problem for the generalized Navier-Stokes equations containing perturbations of retarded type. The degree theory for maps A − g, where A is invertible and g is A-condensing, is used.Various problems for the Navier-Stokes equations describing the motion of the Newton fluid, and its generalizations for nonlinearly-viscous and viscoelastic fluids, have been developed in many papers. We mention here some of the papers which contain surveys on this subject, different approaches, constructions, and methods of investigation:Here we consider the problem of the existence of weak solutions, in the Hopf sense, of the initial-boundary value problem for equations of the NavierStokes type. These equations include the ones describing the movement of nonlinear-viscous and viscous-elastic fluids. We reduce the above problem to an evolution equation in the space of functionals, and then to the equivalent operator equation. The method of this paper consists of constructing operator equations which approximate the original ones, and then investigating their solvability by means of infinite-dimensional degree theory. As we know, the Galerkin-Faedo method or iteration methods have already been used instead of the degree theory for the classical Navier-Stokes equations and for some their generalizations (see, for example, 2 V. T. DMITRIENKO AND V. G. ZVYAGINThis paper consists of four sections.In the first section we introduce the main notations and notions, set up the problem of weak solutions of the initial-boundary value problem for generalized Navier-Stokes equations, and formulate our main results of existence and uniqueness of weak solutions.In the second section the problem of weak solutions is reduced to the investigation of an equivalent operator equation. Then we construct the approximating equations and investigate the properties of the operators involved.In the third section a priori estimates of solutions of approximating equations are established and a proposition on the existence of solutions of such equations is obtained.In the last section the possibility of the limit procedure in the sequence of solutions of approximating equations is established. We present two different approaches to proven the convergence and, as a corollary, we get propositions for the existence of weak solutions of the initial-boundary value problem for some cases of the generalized Navier-Stokes equations. We consider the uniqueness of solutions for dimension n = 2 as well.It should be noted that our interest in this problem arose when Professor P. E. Sobolevskii posed to one of the authors the question of the applicability of topological methods to the initial-boundary value problems in hydrodynamics. The authors are grateful to P. E. Sobolevskii , and Yu. A. Agranovich for discussions on some problems in hydrodynamics.

On weak solutions of the equations of motion of a viscoelastic medium with variable boundary

“…Let γ k be a Kuratovsky's noncompactness measure (see [1]) in E * with the norm · The definition of theL-condensing map is given in [3]. The proof of theorem repeats the proof of Theorem 2.2 in [21] on the strength of Lemmas 5.6-5.8 and the inequality…”

“…in the similar to [3] way. For n ≤ 4 the inclusion V t ⊂ L 4 (Ω t ) is uniformly continuous with respect to t. Then we have by definition of K 3 ε for w ∈ V t the inequality…”

“…It is known that (K 3 ε (w(t)),w(t)) t = 0 for all t ∈ [0,T] (see [3]). Therefore, the integration of both parts of the identity on [0, t] yields…”

“…In the present work, the existence of weak solutions to a regularized initial boundary value problem (1.1) in a domain with a time-dependent boundary Γ t is established. The approximation-topological methods suggested and advanced in [3,4] are used in the paper. It assumes replacement of the problem under consideration by an operator equation, approximation of the equation in a weak sense and application of the topological theory of a degree that allows to establish the existence of solutions on the basis of a priori estimates and statements about passage to the limit.…”

On the existence of weak solutions for the initial‐boundary value problem in the Jeffreys model of motion of a viscoelastic medium

“…We note that in a number of papers (see, e.g., [1,2,6]) the initial-boundary value problem was studied provided that the full derivative d/dt was replaced by the partial derivative ∂/∂t, what essentially narrows the class of mediums satisfying the model [8].…”