A Note on Strong Solutions to the Stokes System (original) (raw)

On the Stokes equations with the Navier-type boundary conditions

Differential Equations & Applications, 2011

In a possibly multiply-connected three dimensional bounded domain, we prove in the L p theory the existence and uniqueness of vector potentials, associated with a divergence-free function and satisfying non homogeneous boundary conditions. Furthermore, we consider the stationary Stokes equations with nonstandard boundary conditions of the form u • n = g and curl u × n = h × n on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions. Our proofs are mainly based on In f − Sup conditions.

On the {\mathscr {}}$$-Bounded Solution Operator and the Maximal L_p$$Lp-$$L_q$$Lq egularity of the Stokes Equations With Free Boundary Condition

2016

In this paper, we consider the boundary value problem of Stokes operator arising in the study of free boundary problem for the Navier-Stokes equations with surface tension in a uniform \(W^{3-1/r}_r\) domain of N-dimensional Euclidean space \({{\mathbb {R}}}^N\) (\(N\geqslant 2\), \(N< r < \infty \)). We prove the existence of \({\mathscr {R}}\)-bounded solution operator with spectral parameter \(\lambda \) varying in a sector \(\varSigma _{\varepsilon , \lambda _0} = \{\lambda \in {{\mathbb {C}}}\mid |\arg \lambda | \leqslant \pi -\varepsilon , \ |\lambda | \geqslant \lambda _0\}\) (\(0< \varepsilon < \pi /2\)), and the maximal \(L_p\)-\(L_q\) regularity with the help of the \({\mathscr {R}}\)-bounded solution operator and the Weis operator valued Fourier multiplier theorem. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely it is assumed the unique existence of solution \({\mathfrak {p}}\in {\mathscr {W}}^1_q(...

A New Proof of the Existence of Suitable Weak Solutions and Other Remarks for the Navier-Stokes Equations

Applied Mathematics, 2018

We prove that the limits of the semi-discrete and the discrete semi-implicit Euler schemes for the 3D Navier-Stokes equations supplemented with Dirichlet boundary conditions are suitable in the sense of Scheffer [1]. This provides a new proof of the existence of suitable weak solutions, first established by Caffarelli, Kohn and Nirenberg [2]. Our results are similar to the main result in [3]. We also present some additional remarks and open questions on suitable solutions.

ON THE UNIQUENESS OF BOUNDED WEAK SOLUTIONS TO THE NAVIER-STOKES CAUCHY PROBLEM

In this note we give a uniqueness theorem for solutions (u, π) to the Navier-Stokes Cauchy problem, assuming that u belongs to L ∞ ((0, T ) × R n ) and (1 + |x|) −n−1 π ∈ L 1 (0, T ; L 1 (R n )), n ≥ 2. The interest to our theorem is motivated by the fact that a possible pressure field π, belonging to L 1 (0, T ; BMO), satisfies in a suitable sense our assumption on the pressure, and by the fact that the proof is very simple.

Weak and Strong Solutions of the Navier-Stokes Initial Value Problem

This paper reviews the existence, uniqueness and regularity of weak and strong solutions of the Navier-Stokes system. For this purpose we emphasize semigroup theory and the theory of the Stokes operator. We use dimensional analysis to clarify the meaning of the results for the solutions. § 0. Introduction Let D be a bounded domain in R" (n>2) with smooth boundary S. We consider the initial-boundary value problem for the Navier-Stokes equations (NS) duldt-Au+(u, grad)w + gradp=/, divw = 0 in Dx(0, T), w=0 on Sx(0, T), M(X, 0) = 0(x) in D, where (u, gi"ad)=£" = 1 u j (d/dXj). This system describes the motion of viscous incompressible fluid filling a rigid vessel D. The function M=(M I (X, f), ..., M"(X, 0) represents the velocity of the fluid and p(x, t) is the pressure. The function a = (a 1 (x),..., a"(x)) is a given initial velocity and /=(/ 1 (x, i) 9 ...,f n (x, 0) is a given external force. We discuss the existence, uniqueness and regularity of weak and strong solutions of this problem. There is an extensive literature on this subject since Leray [27-29] introduced many useful and fundamental ideas. In [29] he constructed a global (in time) weak solution and a local strong solution of the initial value problem when D = R 3. Hopf [20] has proved the existence of a global weak solution of the initial-boundary value problem. Such weak solu

A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W?1/q,q

Mathematische Annalen, 2005

We develop a theory for a general class of very weak solutions to stationary Stokes and Navier-Stokes equations in a bounded domain with boundary ∂ of class C 2,1 , corresponding to boundary data in the distribution space W −1/q,q (∂ ), 1 < q < ∞. These solutions exist and are unique (for small data, in the nonlinear case) in their class of existence, and satisfy a correponding estimate in terms of the data. Moreover, they become regular if the data are regular. To our knowledge, the only existence result for solutions attaining such boundary data is due to Giga, [16], Proposition 2.2, for the Stokes case. However, the methods and the approach used in the present paper are different than Giga's and cover more general issues, including the nonlinear Navier-Stokes equations and the precise way in which the boundary data are attained by the solutions. We also introduce, in the last section, a further generalization of the solution class.

On the maximal LpL_pLp-$L_q$ regularity of the Stokes problem with first order boundary condition; model problems

Journal of the Mathematical Society of Japan, 2012

In this paper, we proved the generalized resolvent estimate and the maximal Lp-Lq regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the R boundedness of solution operators defined in a sector Σ ,γ 0 = {λ ∈ C \ {0} | | arg λ| ≤ π − , |λ| ≥ γ 0 } with 0 < < π/2 and γ 0 ≥ 0. This R boundedness implies the resolvent estimate of the Stokes operator and the combination of this R boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal Lp-Lq regularity of the non-stationary Stokes. For a densely defined closed operator A, we know that what A has maximal Lp regularity implies that the resolvent estimate of A in λ ∈ Σ ,γ 0 , but the opposite direction is not true in general (cf. Kalton and Lancien [19]). However, in this paper using the R boundedness of the operator family in the sector Σ ,λ 0 , we derive a systematic way to prove the resolvent estimate and the maximal Lp regularity at the same time.

On H 2 -Estimates of Solutions to the Stokes System with an Artificial Boundary Condition

Journal of Mathematical Fluid Mechanics, 2002

We consider the Stokes system in a truncated exterior domain, with a local artificial boundary condition on the truncating sphere. The second derivatives of the velocity and the first derivatives of the pressure are estimated in the L 2 -norm, and it is shown how the constants in these estimates depend on the radius of the truncating sphere.

On the existence of local strong solutions for the Navier–Stokes equations in completely general domains

Nonlinear Analysis: Theory, Methods & Applications, 2010

There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains Ω ⊆ R 3 , although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the L q -theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q = 2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value u 0 ∈ L 2 σ (Ω) and a zero external force. Then the condition ∞ 0 e −tA u 0 8 4 dt < ∞ is sufficient and necessary for the existence of a unique local strong solution u ∈ L 8 (0, T ; L 4 (Ω)) in some interval [0, T ), 0 < T ≤ ∞, with u(0) = u 0 , satisfying Serrin's condition 2 8 + 3 4 = 1. Note that Fujita-Kato's sufficient condition u 0 ∈ D(A 1/4 ) is strictly stronger and therefore not optimal.

Generalized Resolvent Estimates of the Stokes Equations with First Order Boundary Condition in a General Domain

Journal of Mathematical Fluid Mechanics, 2013

In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain Ω of the N-dimensional Eulidean space R N , N ≥ 2. This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter λ varying in a sector Σ σ,λ 0 = {λ ∈ C | | arg λ| < π − σ, |λ| ≥ λ 0 }, where 0 < σ < π/2 and λ 0 ≥ 1. The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution p ∈Ŵ 1 q,Γ (Ω) to the variational problem: (∇p, ∇ϕ) = (f, ∇ϕ) for any ϕ ∈Ŵ 1 q ,Γ (Ω). Here, 1 < q < ∞, q = q/(q − 1),Ŵ 1 q,Γ (Ω) is the closure of W 1 q,Γ (Ω) = {p ∈ W 1 q (Ω) | p| Γ = 0} by the semi-norm ∇ • Lq (Ω) , and Γ is the boundary of Ω. In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in (λ 0 , ∞). Our assumption is satisfied for any q ∈ (1, ∞) by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q = 2.