On the reduction of PDE's problems in the half-space, under the slip boundary condition, to the corresponding problems in the whole space (original) (raw)
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Journal of Mathematical Analysis and Applications, 2011
The study of a very large class of linear and non-linear, stationary and evolutive partial differential problems in the half-space (or similar) under the slip boundary condition is reduced here to the much simpler study of the corresponding results for the same problem in the whole space. The approach is particularly suitable for proving new results in strong norms. To determine whether this extension is available, turns out to be a simple exercise. The verification depends on a few general features of the functional space X related to the space variables. Hence, we present an approach as much as possible independent of the particular space X. We appeal to a reflection technique. Hence a crucial assumption is to be in the presence of flat boundaries (see below). Instead of stating "general theorems" we rather prefer to illustrate how to apply our results by considering a couple of interesting problems. As a main example, we show that sharp vanishing viscosity limit results that hold for the evolution Navier-Stokes equations in the whole space can be extended to the slip boundary value problem in the half-space. We also show some applications to non-Newtonian fluid problems.
Communications in Mathematical Physics, 2012
We consider the vanishing-viscosity limit for the Navier-Stokes equations with certain slip-without-friction boundary conditions in a bounded domain with nonflat boundary. In particular, we are able to show convergence in strong norms for a solution starting with initial data belonging to the special subclass of data with vanishing vorticity on the boundary. The proof is obtained by smoothing the initial data and by a perturbation argument with quite precise estimates for the equations of the vorticity and for that of the curl of the vorticity.
This is the second of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space in either 2D or 3D. Under the assumption of analytic initial data, we construct solutions of Navier-Stokes for a short time which is independent of the viscosity. The Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, which were constructed in the first paper, plus an error term. This shows that the Navier-Stokes solution goes to an Euler solution outside a boundary layer and to a solution of the Prandtl equations within the boundary layer. The error term is written as a sum of first order Euler and Prandtl corrections plus a further error term. The equation for the error term is weakly nonlinear; its linear part is the time dependent Stokes equation. This error equation is solved by inversion of the Stokes equation, through expressing the solution as a regular (Euler-like) part plus a boundary layer (Prandtl-like) part. The main technical tool in this analysis is the Abstract Cauchy-Kowalewski Theorem.
Journal of Mathematical Analysis and Applications, 2017
This notes concern the sufficient condition for regularity of solutions to the evolution Navier-Stokes equations known in the literature as Prodi-Serrin's condition. H.-O. Bae and H.-J. Choe proved in a 1999 paper that, in the whole space Ê 3 , it is merely sufficient that two components of the velocity satisfy the above condition. Below, we extend the result to the half-space case Ê n + under slip boundary conditions. We show that it is sufficient that the velocity component parallel to the boundary enjoys the above condition. Flat boundary geometry seems not essential, as suggested by some preliminary calculations in cylindrical domains.
This is the first of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space. In this paper we prove short time existence theorems for the Euler and Prandtl equations with analytic initial data in either two or three spatial dimensions. The main technical tool in this analysis is the abstract Cauchy-Kowalewski theorem. For the Euler equations, the projection method is used in the primitive variables, to which the Cauchy-Kowalewski theorem is directly applicable. For the Prandtl equations, Cauchy-Kowalewski is applicable once the diffusion operator in the vertical direction is inverted.
HAL (Le Centre pour la Communication Scientifique Directe), 2019
In this paper, we study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain Ω ⊂ R 3 of class C 1,1 from the viewpoint of the behavior of solutions with respect to the friction coefficient α. We first prove the existence of a unique weak solution (and strong) in W 1,p (Ω) (and W 2,p (Ω)) of the linear problem for all 1 < p < ∞ considering minimal regularity of α, using some inf-sup condition concerning the rotational operator. Furthermore, we deduce uniform estimates of the solutions for large α which enables us to obtain the strong convergence of Stokes solutions with Navier slip boundary condition to the one with no-slip boundary condition as α → ∞. Finally, we discuss the same questions for the non-linear system. To cite this article: P.
Sharp Inviscid Limit Results under Navier Type Boundary Conditions. An Lp Theory
Journal of Mathematical Fluid Mechanics, 2010
We consider the evolutionary Navier-Stokes equations with a Navier slip-type boundary condition, and study the convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We obtain quite sharp results in the 2-D and 3-D cases. However, in the 3-D case, we need to assume that the boundary is flat. . 35Q30, 76D05, 76D09.
Concerning the W k,p -Inviscid Limit for 3-D Flows Under a Slip Boundary Condition
Journal of Mathematical Fluid Mechanics, 2011
We consider the 3 − D evolutionary Navier-Stokes equations with a Navier slip-type boundary condition, see (1.2), and study the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spaces W k, p (Ω), for arbitrarily large k and p (for previous results see [42] and [9]). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds, as shown below. The cause of this different behavior is quite subtle.
Regularity for the Navier--Stokes equations with slip boundary condition
Proceedings of The American Mathematical Society, 2008
For the Navier-Stokes equations with slip boundary conditions, we obtain the pressure in terms of the velocity. Based on the representation, we consider the relationship in the sense of regularity between the Navier-Stokes equations in the whole space and those in the half space with slip boundary data.