On the Nonhomogeneous Boundary Value Problem for the Navier–Stokes System in a Class of Unbounded Domains

The stationary Navier–Stokes system with nonhomogeneous boundary conditions is studied in a class of domains Ω having “paraboloidal” outlets to infinity. The boundary ${\partial\Omega}$ is multiply connected and consists of M infinite connected components S _m , which form the outer boundary, and I compact connected components Γ _i forming the inner boundary Γ. The boundary value a is assumed to have a compact support and it is supposed that the fluxes of a over the components Γ _i of the inner boundary are sufficiently small. We do not pose any restrictions on fluxes of a over the infinite components S _m . The existence of at least one weak solution to the Navier–Stokes problem is proved. The solution may have finite or infinite Dirichlet integral depending on geometrical properties of outlets to infinity.     

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.     

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.     

A New Class of Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Data

We investigate a class of weak solutions, the so-called very weak solutions, to stationary and nonstationary Navier–Stokes equations in a bounded domain . This notion was introduced by Amann [3], [4] for the nonstationary case with nonhomogeneous boundary data leading to a very large solution class of low regularity. Here we are mainly interested in the investigation of the “largest possible” class of solutions u for the more general problem with arbitrary divergence k = div u, boundary data g = u|_∂Ω and an external force f, as weak as possible, but maintaining uniqueness. In principle, we will follow Amann’s approach.     

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.     

Random Data Cauchy Theory for the Generalized Incompressible Navier–Stokes Equations

In this paper, we consider the generalized Navier?CStokes equations where the space domain is ${\mathbb{T}^N}$ or ${\mathbb{R}^N, N\geq3}$ . The generalized Navier?CStokes equations here refer to the equations obtained by replacing the Laplacian in the classical Navier?CStokes equations by the more general operator (???) ^?? with ${\alpha\in (\frac{1}{2},\frac{N+2}{4})}$ . After a suitable randomization, we obtain the existence and uniqueness of the local mild solution for a large set of the initial data in ${H^s, s\in[-\alpha,0]}$ , if ${1 < \alpha < \frac{N+2}{4}, s\in(1-2\alpha,0]}$ , if ${\frac{1}{2} < \alpha\leq 1}$ . Furthermore, we obtain the probability for the global existence and uniqueness of the solution. Specially, our result shows that, in some sense, the Cauchy problem of the classical Navier?CStokes equation is local well-posed for a large set of the initial data in H ^?1+, exhibiting a gain of ${\frac{N}{2}-}$ derivatives with respect to the critical Hilbert space ${H^{\frac{N}{2}-1}}$ .     

Global Wellposedness for the 3D Inhomogeneous Incompressible Navier–Stokes Equations

This paper addresses the three-dimensional Navier–Stokes equations for an incompressible fluid whose density is permitted to be inhomogeneous. We establish a theorem of global existence and uniqueness of strong solutions for initial data with small ${\dot{H}^{\frac12}}$ -norm, which also satisfies a natural compatibility condition. A key point of the theorem is that the initial density need not be strictly positive.     

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.     

Stability Properties of POD–Galerkin Approximations for the Compressible Navier–Stokes Equations

Fluid flows are very often governed by the dynamics of a mall number of coherent structures, i.e., fluid features which keep their individuality during the evolution of the flow. The purpose of this paper is to study a low order simulation of the Navier–Stokes equations on the basis of the evolution of such coherent structures. One way to extract some basis functions which can be interpreted as coherent structures from flow simulations is by Proper Orthogonal Decomposition (POD). Then, by means of a Galerkin projection, it is possible to find the system of ODEs which approximates the problem in the finite-dimensional space spanned by the POD basis functions. It is found that low order modeling of relatively complex flow simulations, such as laminar vortex shedding from an airfoil at incidence and turbulent vortex shedding from a square cylinder, provides good qualitative results compared with reference computations. In this respect, it is shown that the accuracy of numerical schemes based on simple Galerkin projection is insufficient and numerical stabilization is needed. To conclude, we approach the issue of the optimal selection of the norm, namely the H ¹ norm, used in POD for the compressible Navier–Stokes equations by several numerical tests. Received 21 April 1999 and accepted 18 November 1999     

On Kato’s Method for Navier–Stokes Equations

We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier–Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in \mathbbR³{\mathbb{R}}^{3} and irregular domains in \mathbbRⁿ{\mathbb{R}}^{n}.     

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 ${{K=K(\Omega, q)>0}}$ .     

Ill-posedness of the Hydrostatic Euler and Navier–Stokes Equations

We prove that the linearization of the hydrostatic Euler equations at certain parallel shear flows is ill-posed. The result also extends to the hydrostatic Navier–Stokes equations with a small viscosity.     

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.     

Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations

For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.     

Mixed Boundary Value Problems for the Stationary Navier–Stokes System in Polyhedral Domains

The authors consider boundary value problems for the Navier–Stokes system in a polyhedral domain, where different boundary conditions (in particular, Dirichlet, Neumann, slip conditions) are arbitrarily combined on the faces of the polyhedron. They prove existence and regularity theorems for weak solutions in weighted (and nonweighted) L _p Sobolev and Hölder spaces with sharp integrability and smoothness parameters.     

The Numerical Technique for the Landslide Tsunami Simulations Based on Navier–Stokes Equations

The paper presents an integral technique simulating all phases of a landslide-driven tsunami. The technique is based on the numerical solution of the system of Navier–Stokes equations for multiphase flows. The numerical algorithm uses a fully implicit approximation method, in which the equations of continuity and momentum conservation are coupled through implicit summands of pressure gradient and mass flow. The method we propose removes severe restrictions on the time step and allows simulation of tsunami propagation to arbitrarily large distances. The landslide origin is simulated as an individual phase being a Newtonian fluid with its own density and viscosity and separated from the water and air phases by an interface. The basic formulas of equation discretization and expressions for coefficients are presented, and the main steps of the computation procedure are described in the paper. To enable simulations of tsunami propagation across wide water areas, we propose a parallel algorithm of the technique implementation, which employs an algebraic multigrid method. The implementation of the multigrid method is based on the global level and cascade collection algorithms that impose no limitations on the paralleling scale and make this technique applicable to petascale systems. We demonstrate the possibility of simulating all phases of a landslide-driven tsunami, including its generation, propagation and uprush. The technique has been verified against the problems supported by experimental data. The paper describes the mechanism of incorporating bathymetric data to simulate tsunamis in real water areas of the world ocean. Results of comparison with the nonlinear dispersion theory, which has demonstrated good agreement, are presented for the case of a historical tsunami of volcanic origin on the Montserrat Island in the Caribbean Sea.     

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.      

On the Local Smoothness of Solutions of the Navier–Stokes Equations

We consider the Cauchy problem for incompressible Navier–Stokes equations with initial data in , and study in some detail the smoothing effect of the equation. We prove that for T < ∞ and for any positive integers n and m we have , as long as stays finite.