Non‐standard Stokes and Navier–Stokes problems: existence and regularity in stationary case

Abstract

SUMMARYThis paper is devoted to Stokes and Navier-Stokes problems with non-standard boundary conditions: we consider, in particular, the case where the pressure is given on a part of the boundary. These problems were studied by BÃ egue, Conca, Murat and Pironneau. They proved the existence of variational solutions, indicating that these were solutions of the initial non-standard problems, if they are regular enough, but without specifying the conditions on the data which would imply this regularity. In this paper, ÿrst we show that the variational solutions, on supposing pressure on the boundary 2 of regularity H 1=2 instead of H −1=2 , have their Laplacians in L 2 and, therefore, are solutions of non-standard Stokes problem. Next, we give a result of regularity H 2 , which we generalize, obtaining regularities W m; r , m ∈ N; m¿2; r¿2. Finally, by a ÿxed-point argument, we prove analogous results for the Navier-Stokes problem, in the case where the viscosity is large compared to the data.

Some properties on the surfaces of vector fields and its application to the Stokes and Navier–Stokes problems with mixed boundary conditions

“…In this case equivalence between the norms ∥v∥ H 1 (Ω) and ∥rot v∥ L 2 (Ω) under some conditions (see Theorem A.1 in [31], Lemma 2 in [49]) is on the base and in this way many problems are studied (cf. [1,6,12,13,[23][24][25]29,31,32,49,58,59]). In Section 1 of [29], the bilinear form (1.5) instead of (1.7) is used since two bilinear forms (1.5) and (1.7) for polygon or polyhedral domain under some boundary conditions are equal (cf.…”

“…Also, on a part of boundary one deals with the total pressure (Bernoulli's pressure) 1 2 |v| 2 + p (cf. [23,24]) or static pressure p (cf. [4,59]).…”

A projection method for Navier-Stokes equations with a boundary condition including the total pressure

“…We will show the existence in a different way (Corollary 2.24). The stationary case has been studied in [3,4,7,12,25]. In [7,8], the finite element discretization problems with this type of boundary condition are proposed.…”

Some Remarks on the Boundary Conditions in the Theory of Navier-Stokes Equations

“…The situation of bounded domains, eventually multiply-connected with boundary not connected, has been investigated by Begue-Conca-Murat-Pironneau [8] for the linear and nonlinear cases (see also Ebmeyer-Frehse [21] for some mixed boundary conditions in polyhedral domains). Next, Bernard [12]- [13] and more recently Kozono-Yanagisawa [30] and Amrouche-Seloula [5] and [4] completed this study by developing very useful properties concerning the vector potentials, some Sobolev inequalities for vector fields and a complete L p -theory to solve Stokes equations with boundary condition (1.6) or (1.8).…”