Stokes and Navier–Stokes Equations with Robin Boundary Conditions in a Half-Space

We study the initial-boundary value problem for the Stokes equations with Robin boundary conditions in the half-space It is proved that the associated Stokes operator is sectorial and admits a bounded H^∞-calculus on As an application we prove also a local existence result for the nonlinear initial value problem of the Navier–Stokes equations with Robin boundary conditions.     

Weak Solutions of the Navier–Stokes Equations for Compressible Flows in a Half-Space with No-Slip Boundary Conditions

We consider the Navier–Stokes equations for the motion of compressible, viscous flows in a half-space ${\mathbb{R}^n_+,}$ n =  2,  3, with the no-slip boundary conditions. We prove the existence of a global weak solution when the initial data are close to a static equilibrium. The density of the weak solution is uniformly bounded and does not contain a vacuum, the velocity is Hölder continuous in (x, t) and the material acceleration is weakly differentiable. The weak solutions of this type were introduced by D. Hoff in Arch Ration Mech Anal 114(1):15–46, (1991), Commun Pure and Appl Math 55(11):1365–1407, (2002) for the initial-boundary value problem in ${\Omega = \mathbb{R}^n}$ and for the problem in ${\Omega = \mathbb{R}^n_+}$ with the Navier boundary conditions.     

2D Navier–Stokes Equations in a Time Dependent Domain with Neumann Type Boundary Conditions

In this paper the two-dimensional Navier–Stokes system for incompressible fluid coupled with a parabolic equation through the Neumann type boundary condition for the second component of the velocity is considered. Navier–Stokes equations are defined on a given time dependent domain. We prove the existence of a weak solution for this system. In addition, we prove the continuous dependence of solutions on the data for a regularized version of this system. For a special case of this regularized system also a problem with an unknown interface is solved.     

On 3D Lagrangian Navier–Stokes α Model with a Class of Vorticity-Slip Boundary Conditions

This paper concerns the 3-dimensional Lagrangian Navier–Stokes α model and the limiting Navier–Stokes system on smooth bounded domains with a class of vorticity-slip boundary conditions and the Navier-slip boundary conditions. It establishes the spectrum properties and regularity estimates of the associated Stokes operators, the local well-posedness of the strong solution and global existence of weak solutions for initial boundary value problems for such systems. Furthermore, the vanishing α limit to a weak solution of the corresponding initial-boundary value problem of the Navier–Stokes system is proved and a rate of convergence is shown for the strong solution.     

Uniform Regularity for the Navier–Stokes Equation with Navier Boundary Condition

We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier–Stokes equation with the Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in L ^∞. This allows us to obtain the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.     

On the Flux Problem in the Theory of Steady Navier–Stokes Equations with Nonhomogeneous Boundary Conditions

We study the nonhomogeneous boundary value problem for Navier–Stokes equations of steady motion of a viscous incompressible fluid in a two-dimensional, bounded, multiply connected domain ${\Omega = \Omega_1 \backslash \overline{\Omega}_2, \overline\Omega_2\subset \Omega_1}$ . We prove that this problem has a solution if the flux ${\mathcal{F}}$ of the boundary value through ?Ω ₂ is nonnegative (inflow condition). The proof of the main result uses the Bernoulli law for a weak solution to the Euler equations and the one-sided maximum principle for the total head pressure corresponding to this solution.     

Vanishing Viscous Limits for 3D Navier–Stokes Equations with a Navier-Slip Boundary Condition

In this paper, we investigate the vanishing viscosity limit for solutions to the Navier–Stokes equations with a Navier slip boundary condition on general compact and smooth domains in R ³. We first obtain the higher order regularity estimates for the solutions to Prandtl’s equation boundary layers. Furthermore, we prove that the strong solution to Navier–Stokes equations converges to the Eulerian one in C([0, T]; H ¹(Ω)) and ${L^\infty((0,T) \times \Omega)}$ , where T is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also.     

Navier–Stokes Equations with Shear-Thickening Viscosity. Regularity up to the Boundary

In this article we prove some sharp regularity results for the stationary and the evolution Navier–Stokes equations with shear dependent viscosity, see (1.1), under the no-slip boundary condition(1.4). We are interested in regularity results for the second order derivatives of the velocity and for the first order derivatives of the pressure up to the boundary, in dimension n ≥ 3. In reference [4] we consider the stationary problem in the half space \mathbbR₊ⁿ{\mathbb{R}}_+^n under slip and no-slip boundary conditions. Here, by working in a simpler context, we concentrate on the basic ideas of proofs. We consider a cubic domain and impose our boundary condition (1.4) only on two opposite faces. On the other faces we assume periodicity, as a device to avoid unessential technical difficulties. This choice is made so that we work in a bounded domain Ω and, at the same time, with a flat boundary. In the last section we provide the extension of the results from the stationary to the evolution problem.     

Besov Space Regularity Conditions for Weak Solutions of the Navier–Stokes Equations

Consider a bounded domain ${{\Omega \subseteq \mathbb{R}^3}}$ with smooth boundary, some initial value ${{u_0 \in L^2_{\sigma}(\Omega )}}$ , and a weak solution u of the Navier–Stokes system in ${{[0,T) \times\Omega,\,0 < T \le \infty}}$ . Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space $$B^{q,s}(\Omega ):=\left\{v\in L^2_{\sigma}(\Omega ); \|v\|_{B^{q,s}(\Omega )} := \left(\int\limits^{\infty}_0 \left\|e^{-\tau A}v\right\|^s_q {\rm d} \tau\right)^{1/s}<\infty \right\}$$ with ${{2 < s < \infty,\,3 < q <\infty,\,\frac2{s}+\frac{3}{q} = 1}}$ ; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012), is a subspace of the well known Besov space ${{{\mathbb{B}}^{-2/s}_{q,s}(\Omega )}}$ , see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002). Our main results on the regularity of u exploits a variant of the space ${{B^{q,s}(\Omega )}}$ in which the integral in time has to be considered only on finite intervals (0, δ ) with ${{\delta \to 0}}$ . Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition ${{u\in L^{\infty}_{\text{loc}}([0,T);L^3_{\sigma}(\Omega ))}}$ . Finally, we obtain a large class of regular weak solutions u defined by a smallness condition ${{\|u_0\|_{B^{q,s}(\Omega )} \le K}}$ with some constant ${{K=K(\Omega, q)>0}}$ .     

Navier–Stokes Equations with Shear Thinning Viscosity. Regularity up to the Boundary

In this paper we consider a class of stationary Navier–Stokes equations with shear dependent viscosity, in the shear thinning case p < 2, under a non-slip boundary condition. We are interested in global (i.e., up to the boundary) regularity results, in dimension n = 3, for the second order derivatives of the velocity and the first order derivatives of the pressure. As far as we know, there are no previous global regularity results for the second order derivatives of the solution to the above boundary value problem. We consider a cubic domain and impose the non-slip boundary condition only on two opposite faces. On the other faces we assume periodicity, as a device to avoid effective boundary conditions. This choice is made so that we work in a bounded domain Ω and simultaneously with a flat boundary. The extension to non-flat boundaries is done in the forthcoming paper [7], by following ideas introduced by the author, for the case p > 2, in reference [5]. The results also hold in the presence of the classical convective term, provided that p is sufficiently close to the value 2.      

New Sufficient Conditions of Local Regularity for Solutions to the Navier–Stokes Equations

New sufficient conditions of local regularity for suitable weak solutions to the non-stationary three-dimensional Navier–Stokes equations are proved. They contain the celebrated Caffarelli–Kohn–Nirenberg theorem as a particular case.      

Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case of a bounded open set, connected of class C^1,1{\mathcal{C}^{1,1}} of \mathbbR³{\mathbb{R}^3}. Taking up once again the duality method introduced by Lions and Magenes (Problèmes aus limites non-homogènes et applications, vols. 1 & 2, Dunod, Paris, 1968) and Giga (Math Z 178:287–329, 1981) for open sets of class C^￥{\mathcal{C}^{\infty}} [see also chapter 4 of Necas (Les méthodes directes en théorie des équations elliptiques. (French) Masson et Cie, éd., Paris; Academia, éditeurs, Prague, 1967), which considers the Hilbertian case p = 2 for general elliptic operators], we give a simpler proof of the existence of a very weak solution for stationary Oseen and Navier–Stokes equations when data are not regular enough, based on density arguments and a functional framework adequate for defining more rigourously the traces of non-regular vector fields. In the stationary Navier–Stokes case, the results will be valid for external forces not necessarily small, which lets us extend the uniqueness class of solutions for these equations. Considering more regular data, regularity results in fractional Sobolev spaces will also be discussed for the three systems. All these results can be extended to other dimensions.     

Extensions of Serrin’s Uniqueness and Regularity Conditions for the Navier–Stokes Equations

Consider a smooth bounded domain ${\Omega \subseteq {\mathbb{R}}^3}$ , a time interval [0, T), 0?<?T?≤?∞, and a weak solution u of the Navier–Stokes system. Our aim is to develop several new sufficient conditions on u yielding uniqueness and/or regularity. Based on semigroup properties of the Stokes operator we obtain that the local left-hand Serrin condition for each ${t\in (0,T)}$ is sufficient for the regularity of u. Somehow optimal conditions are obtained in terms of Besov spaces. In particular we obtain such properties under the limiting Serrin condition ${u \in L_{\rm loc}^\infty([0,T);L^3(\Omega))}$ . The complete regularity under this condition has been shown recently for bounded domains using some additional assumptions in particular on the pressure. Our result avoids such assumptions but yields global uniqueness and the right-hand regularity at each time when ${u \in L_{\rm loc}^\infty([0,T);L^3(\Omega))}$ or when ${u(t)\in L^3(\Omega)}$ pointwise and u satisfies the energy equality. In the last section we obtain uniqueness and right-hand regularity for completely general domains.     

Weak Solutions and Attractors for Three-Dimensional Navier–Stokes Equations with Nonregular Force

A three-dimensional Navier–Stokes equation is considered. The forcing term is the derivative of a continuous function; the case of white noise is also considered. The aim is to prove the existence of weak solutions and to construct an attractor for the corresponding shift dynamical system in path space, following an idea of Sell.     

Navier�CStokes Equations with Nonhomogeneous Boundary Conditions in a Bounded Three-Dimensional Domain

Following the ideas developed in Girinon (Annales de l’Institut Poincaré. Analyse Non Linéaire 26:2025–2053, 2009), we prove the existence of a weak solution to Navier–Stokes equations describing the isentropic flow of a gas in a bounded region, W ì R³{\Omega\subset \mathbf{R}^{3}} , with nonhomogeneous Dirichlet boundary conditions on ∂Ω.     

About the Regularized Navier–Stokes Equations

The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier–Stokes system. The Marcinkiewicz space L^3, is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical regularized Navier–Stokes systems. The first one was introduced by J. Leray and consists in mollifying the nonlinearity. The second one was proposed by J.-L. Lions, who added the artificial hyper-viscosity (–)^{/ 2}, > 2 to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as t toward solutions of the original Navier–Stokes system.