Initial boundary value problems for viscoelastic Jeffreys fluids

Abstract

We study the initial boundary value problem for the nonlinear system, which describes the dynamics of an incompressible viscoelastic fluid with the Jeffreys constitutive law under the Navier slip boundary condition. We construct a global (in time) weak solution to this problem.

Towards a Proof of Bahri–Coron’s Type Theorem for Mixed Boundary Value Problems

“…Due to the compactness defect of the Sobolev embedding V(Ω) → L 2 * (Ω), the functional J fails to satisfy the Palais-Smale condition on Σ + . In order to describe the sequences failing the Palais-Smale condition, we introduce in the following a family of "almost solutions" of problem (1). For any a ∈ Ω ∪ Γ 1 , and λ > 0, we define…”

“…Elliptic equations involving Laplacian operator on bounded domains with mixed boundary conditions arise in real applications, for example, in hydrodynamics; see [1,2]. Generally, nonlinear problems subject to various boundary conditions appear in many different branches of the applied sciences, including physics (e.g., steady-state heat flux modeling), chemistry (e.g., Keller-Segel model for parabolic equations in chemotaxis), biology (e.g., Gierer-Meinhardt system in the formation of biological models), and engineering.…”

“…The existence of a bounded domain for which S(Ω) is achieved is strange and makes the study of mixed boundary problems of type (1) different compared with the classical homogeneous Dirichlet problem. For more conditions that ensure S(Ω) is achieved for problems such as (1), we refer to [8][9][10][11][12].…”

“…Problem (1) does not always have a solution. A necessary condition to obtain a solution has been established in [6].…”

Global well-posedness for 2-D viscoelastic fluid model

“…Various well-posedness results for motion equations with classical boundary conditions were obtained also in [1], [5], [9], [12]. Note also that slip problems for Oldroyd type models have recently attracted considerable interest (see [3], [4], [6], [7], [14]). A review of mathematical results on viscoelastic fluids models and many references may be found in [15].…”