Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

Periodic solutions of the Navier–Stokes equations with inhomogeneous boundary conditions

We show the existence of time periodic solutions of the Navier–Stokes equations in bounded domains of \mathbb R³{\mathbb R^3} with inhomogeneous boundary conditions in the strong and weak sense. In particular, for weak solutions, we deal with more generalized conditions on the boundary data for Leray’s problem.     

On uniqueness of stationary solutions to the Navier–Stokes equations in exterior domains

The aim of this paper is to prove a uniqueness criterion for solutions to the stationary Navier–Stokes equation in 3-dimensional exterior domains within the class $u\in L_{3,\infty }$ with $?u\in L_{3/2,\infty }$, where $L_{3,\infty }$ and $L_{3/2,\infty }$ are the Lorentz spaces. Our criterion asserts that if $u$ and $v$ are the solutions, $u$ is small in $L_{3,\infty }$ and $u,v\in L_p$ for some $p>3$, then $u=v$. The proof is based on analysis of the dual equation with the aid of the bootstrap argument.     

Weak solutions to the generalized Navier–Stokes equations with mixed boundary conditions and implicit obstacle constraints

This paper is concerned with the investigation of a generalized Navier–Stokes equation for non-Newtonian fluids of Bingham-type (GNSE, for short) involving a multivalued and nonmonotone slip boundary condition formulated by the generalized Clarke subdifferential of a locally Lipschitz superpotential, a no leak boundary condition, and an implicit obstacle inequality. We obtain the weak formulation of (GNSE) which is a generalized quasi-variational–hemivariational inequality. By introducing an Oseen model as an auxiliary (intermediated) problem and employing Kakutani-Ky Fan theorem for multivalued operators as well as the theory of nonsmooth analysis, an existence theorem to (GNSE) is established.     

Existence of global strong solutions for Navier–Stokes equations with external force

We consider the initial boundary value problems (IBVPs) for 1D isentropic compressible Navier–Stokes equations with density-dependent viscosity coefficients and external force. If the initial data is regular, the existence and uniqueness of global strong solution to IBVP are proved in this article.     

Derivation of Prandtl boundary layer equations for the incompressible Navier–Stokes equations in a curved domain

The proposal of this note is to derive the equations of boundary layers in the small viscosity limit for the two-dimensional incompressible Navier–Stokes equations defined in a curved bounded domain with the non-slip boundary condition. By using curvilinear coordinate system in a neighborhood of boundary, and the multi-scale analysis we deduce that the leading profiles of boundary layers of the incompressible flows in a bounded domain still satisfy the classical Prandtl equations when the viscosity goes to zero, which are the same as for the flows defined in the half space.     

Spatially homogeneous solutions of 3D stochastic Navier–Stokes equations and local energy inequality

We study the three-dimensional stochastic Navier–Stokes equations with additive white noise, in the context of spatially homogeneous solutions in R³${\mathbb{R}}^3$, i.e. solutions with a law invariant under space translations. We prove the existence of such a solution, with the additional property of being suitable in the sense of Caffarelli, Kohn and Nirenberg: it satisfies a localized version of the energy inequality.     

Necessary and sufficient conditions for local strong solvability of the Navier–Stokes equations in exterior domains

We consider the instationary Navier–Stokes equations in a smooth exterior domain \({\Omega \subseteq \mathbb{R}^3}\) with initial value u ₀, external force f = div  F and viscosity ν. It is an important question to characterize the class of initial values \({u_0\in L^2_{\sigma}(\Omega)}\) that allow a strong solution \({u \in L^s(0,T; L^q(\Omega))}\) in some interval \({[0,T[ \, , 0 < T \leq \infty}\) where s, q with 3 < q < ∞ and \({\frac{2}{s} + \frac{3}{q} =1}\) are so-called Serrin exponents. In Farwig and Komo (Analysis (Munich) 33:101–119, 2013) it is proved that \({\int_0^{\infty} \| e^{-\nu t A} u_0 \|_q^{s} \, {d}t < \infty}\) is necessary and sufficient for the existence of a strong solution \({u \in L^s(0,T ; L^q(\Omega)) \, , 0 < T \leq \infty}\) , if additionally 3 < q ≤ 8; here, A denotes the Stokes operator. In this paper, we will show that this result remains true if q > 8, and consequently, \({\int_0^{\infty} \| e^{-\nu t A} u_0 \|_q^{s} \, {d}t < \infty}\) is the optimal initial value condition to obtain such a strong solution for all possible Serrin exponents s, q.     

On weak D-solutions to the non-stationary Navier–Stokes equations in a three-dimensional exterior domain

In this note we study the Navier–Stokes initial boundary value problem in exterior domains. We assume that the initial data has just finite Dirichlet norm. We call the solution \(D\) -solution. It is well known that the analogous steady problem is solved in Galdi (An Introduction to the Mathematical Theory of the Navier–Stokes Equations II. Springer, Berlin, 1994), as well as the existence of time periodic solutions in Maremonti et al. (J Math Sci 93(5):719–746, 1999, Zap. Nauchn. Semin. POMI 233:142–182, 1996). So it is natural to inquire about the case of the nonstationary problem.     

Basic properties of solution of the non-steady Navier–Stokes equations with mixed boundary conditions in a bounded domain

In this paper we deal with the system of the non-steady Navier–Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. Suppose that the system is solvable with some given data (the initial velocity and the right hand side). We prove that there exists a unique solution of this system for data which are small perturbations of the previous ones.     

Global well-posedness of strong solutions to the two-dimensional barotropic compressible Navier–Stokes equations with vacuum

The aim of this paper is to establish the global well-posedness and large-time asymptotic behavior of the strong solution to the Cauchy problem of the two-dimensional compressible Navier–Stokes equations with vacuum. It is proved that if the shear viscosity \({\mu}\) is a positive constant and the bulk viscosity \({\lambda}\) is the power function of the density, that is, \({\lambda=\rho^{\beta}}\) with \({\beta \in [0,1],}\) then the Cauchy problem of the two-dimensional compressible Navier–Stokes equations admits a unique global strong solution provided that the initial data are of small total energy. This result can be regarded as the extension of the well-posedness theory of classical compressible Navier–Stokes equations [such as Huang et al. (Commun Pure Appl Math 65:549–585, 2012) and Li and Xin (http://arxiv.org/abs/1310.1673) respectively]. Furthermore, the large-time behavior of the strong solution to the Cauchy problem of the two-dimensional barotropic compressible Navier–Stokes equations had been also obtained.     

Solutions of the Navier–Stokes equations with various types of boundary conditions

In this paper we consider the initial boundary value problem of the Navier–Stokes system with various types of boundary conditions. We study the global-in-time existence and uniqueness of a solution of this system. In particular, suppose that the problem is solvable with some given data (the initial velocity and the external body force). We prove that there exists a unique solution for data which are small perturbations of the previous ones.     

Global existence of a weak solution to 3d stochastic Navier–Stokes equations in an exterior domain

In this paper we consider the existence of a weak solution to a 3d stochastic Navier–Stokes equation perturbed by a noise g(X(t))dW, where W(t) is a cylindrical Wiener process, in an exterior domain D: $$dX(t) = [-AX(t) + B\left( X(t)\right)]dt + g(X(t))dW(t),$$ where \({A = -P_{2}\Delta}\) is the Stokes operator and g satisfies some conditions.