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.      

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.     

Navier–Stokes Equations with Navier Boundary Conditions for an Oceanic Model

We consider the Navier–Stokes equations in a thin domain of which the top and bottom surfaces are not flat. The velocity fields are subject to the Navier conditions on those boundaries and the periodicity condition on the other sides of the domain. This toy model arises from studies of climate and oceanic flows. We show that the strong solutions exist for all time provided the initial data belong to a “large” set in the Sobolev space H ¹. Furthermore we show, for both the autonomous and the nonautonomous problems, the existence of a global attractor for the class of all strong solutions. This attractor is proved to be also the global attractor for the Leray–Hopf weak solutions of the Navier–Stokes equations. One issue that arises here is a nontrivial contribution due to the boundary terms. We show how the boundary conditions imposed on the velocity fields affect the estimates of the Stokes operator and the (nonlinear) inertial term in the Navier–Stokes equations. This results in a new estimate of the trilinear term, which in turn permits a short and simple proof of the existence of strong solutions for all time.     

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.     

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.      

Anisotropic Regularity Conditions for the Suitable Weak Solutions to the 3D Navier–Stokes Equations

We are concerned with the problem, originated from Seregin (159–200, 2007), Seregin (J. Math. Sci. 143: 2961–2968, 2007), Seregin (Russ. Math. Surv. 62:149–168, 2007), what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3D Navier–Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled L ² norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions.     

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.     

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 ${{K=K(\Omega, q)>0}}$ .     

Vanishing Viscosity Limit of the Compressible Navier–Stokes Equations for Solutions to a Riemann Problem

We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical nonlinear hyperbolic waves.     

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.     

A Weak-L p Prodi–Serrin Type Regularity Criterion for the Navier–Stokes Equations

We give simple proofs that a weak solution u of the Navier–Stokes equations with H ¹ initial data remains strong on the time interval [0, T] if it satisfies the Prodi–Serrin type condition u ∈ L ^s (0, T;L ^r,∞(Ω)) or if its L ^s,∞(0, T;L ^r,∞(Ω)) norm is sufficiently small, where 3 < r ≤ ∞ and (3/r) + (2/s) = 1.     

Existence and Regularity of Very Weak Solutions of the Stationary Navier–Stokes Equations

We consider the stationary Navier–Stokes equations in a bounded domain Ω in R ⁿ with smooth connected boundary, where n = 2, 3 or 4. In case that n = 3 or 4, existence of very weak solutions in L ⁿ (Ω) is proved for the data belonging to some Sobolev spaces of negative order. Moreover we obtain complete L ^q -regularity results on very weak solutions in L ⁿ (Ω). If n = 2, then similar results are also proved for very weak solutions in with any q ₀ > 2. We impose neither smallness conditions on the external force nor boundary data for our existence and regularity results.     

A Refinement of the Local Serrin-Type Regularity Criterion for a Suitable Weak Solution to the Navier–Stokes Equations

We formulate a new criterion for regularity of a suitable weak solution v to the Navier–Stokes equations at the space-time point (x ₀, t ₀). The criterion imposes a Serrin-type integrability condition on v only in a backward neighbourhood of (x ₀, t ₀), intersected with the exterior of a certain space-time paraboloid with vertex at point (x ₀, t ₀). We make no special assumptions on the solution in the interior of the paraboloid.     

Regularity Criteria of Weak Solutions in terms of the Pressure in Lorentz Spaces to the Navier–Stokes Equations

We consider the incompressible Navier–Stokes equations in Ω ×?(0, T), where Ω is a domain in ${\mathbb{R}^3}$ . We give regularity criteria in terms of the pressure in Lorentz spaces with the corresponding small norm. In particular, our results extend previous ones to the Lorentz space with respect to temporal variable.     

On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z₀=(x₀, t₀)z_{0}=(x_{0}, t_{0}), and let Q_z0,r = B_x0,r ×(t₀ -r², t₀)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.     

On the Two-Phase Navier–Stokes Equations with Boussinesq–Scriven Surface Fluid

Two-phase flows with interface modeled as a Boussinesq–Scriven surface fluid are analysed concerning their fundamental mathematical properties. This extended form of the common sharp-interface model for two-phase flows includes both surface tension and surface viscosity. For this system of partial differential equations with free interface it is shown that the energy serves as a strict Ljapunov functional, where the equilibria of the model without boundary contact consist of zero velocity and spheres for the dispersed phase. The linearizations of the problem are derived formally, showing that equilibria are linearly stable, but nonzero velocities may lead to problems which linearly are not well-posed. This phenomenon does not occur in absence of surface viscosity. The present paper aims at initiating a rigorous mathematical study of two-phase flows with surface viscosity.     

On Nonexistence for Stationary Solutions to the Navier–Stokes Equations with a Linear Strain

We consider stationary solutions to the three-dimensional Navier–Stokes equations for viscous incompressible flows in the presence of a linear strain. For certain class of strains we prove a Liouville type theorem under suitable decay conditions on vorticity fields.     

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.     

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.