Compressible Navier–Stokes Equations on Thin Domains

We consider the barotropic Navier–Stokes system describing the motion of a compressible viscous fluid confined to a straight layer \({\Omega_\varepsilon = \omega\times (0, \varepsilon)}\) , where ω is a particular 2-D domain (a periodic cell, bounded domain or the whole 2-D space). We show that the weak solutions in the 3D domain converge to a (strong) solutions of the 2-D Navier–Stokes system on ω as \({\varepsilon \to 0}\) on the maximal life time of the strong solution.     

Large Time Behavior of Density-Dependent Incompressible Navier–Stokes Equations on Bounded Domains

The large-time asymptotic behavior of classical solutions to the density-dependent incompressible Navier–Stokes equations driven by an external force on bounded domains in 2-D is studied. It is shown that the velocity field and its first-order derivatives converge to zero as time goes to infinity for large initial data and external forces.     

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.     

Global Well-Posedness of 2D Compressible Navier–Stokes Equations with Large Data and Vacuum

In this paper, we study the global well-posedness of the 2D compressible Navier–Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity μ is a positive constant and the bulk viscosity λ is the power function of the density, that is, λ(ρ) = ρ ^β with β > 3, then the 2D compressible Navier–Stokes equations with the periodic boundary conditions on the torus \({\mathbb{T}^2}\) admit a unique global classical solution (ρ, u) which may contain vacuums in an open set of \({\mathbb{T}^2}\) . Note that the initial data can be arbitrarily large to contain vacuum states.     

Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains

Concerning to the non-stationary Navier–Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier–Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7:339–367, 2005), in L ^p spaces for p ≥ 3. In this article, we first extend their result to the case \frac32 < p{\frac{3}{2} < p} by modifying the method in Bae and Jin (J Math Fluid Mech 10:423–433, 2008) that was used to obtain weighted estimates for the Navier–Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier–Stokes flow with a nonzero velocity at infinity.     

Some Uniform Estimates and Large-Time Behavior of Solutions to One-Dimensional Compressible Navier–Stokes System in Unbounded Domains with Large Data

We study the large-time behavior of solutions to the initial and initial boundary value problems with large initial data for the compressible Navier–Stokes system describing the one-dimensional motion of a viscous heat-conducting perfect polytropic gas in unbounded domains. The temperature is proved to be bounded from below and above, independent of both time and space. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity. Note that the initial data can be arbitrarily large. This result is proved by using elementary energy methods.     

A Variational Approach for the Navier–Stokes System

We introduce a variational approach to treat the regularity of the Navier–Stokes equations both in dimensions 2 and 3. Though the method allows the full treatment in dimension 2, we seek to precisely stress where it breaks down for dimension 3. The basic feature of the procedure is to look directly for strong solutions, by minimizing a suitable error functional that measures the departure of feasible fields from being a solution of the problem. By considering the divergence-free property as part of feasibility, we are able to avoid the explicit analysis of the pressure. Two main points in our analysis are:

Non-autonomous 2D Navier–Stokes System with Singularly Oscillating External Force and its Global Attractor

We study the global attractor of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form . If the functions g ₀(x, t) and g ₁ (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor is bounded in the space H, however, its norm may be unbounded as since the magnitude of the external force is growing. Assuming that the function g ₁ (z, t) has a divergence representation of the form where the functions (see Section 3), we prove that the global attractors of the N.–S. equations are uniformly bounded with respect to for all . We also consider the “limiting” 2D N.–S. system with external force g ₀(x, t). We have found an estimate for the deviation of a solution of the original N.–S. system from a solution u ⁰(x, t) of the “limiting” N.–S. system with the same initial data. If the function g ₁ (z, t) admits the divergence representation, the functions g ₀(x, t) and g ₁ (z, t) are translation compact in the corresponding spaces, and , then we prove that the global attractors converges to the global attractor of the “limiting” system as in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of from of the form: in the case, when the global attractor is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).      

Well-Posedness of the Multidimensional Fractional Stochastic Navier–Stokes Equations on the Torus and on Bounded Domains

In this work, we introduce and study the well-posedness of the multidimensional fractional stochastic Navier–Stokes equations on bounded domains and on the torus (briefly dD-FSNSE). For the subcritical regime, we establish thresholds for which a maximal local mild solution exists and satisfies required space and time regularities. We prove that under conditions of Beale–Kato–Majda type, these solutions are global and unique. These conditions are automatically satisfied for the 2D-FSNSE on the torus if the initial data has H ¹-regularity and the diffusion term satisfies growth and Lipschitz conditions corresponding to H ¹-spaces. The case of 2D-FSNSE on the torus is studied separately. In particular, we established thresholds for the global existence, uniqueness, space and time regularities of the weak (strong in probability) solutions in the subcritical regime. For the general regime, we prove the existence of a martingale solution and we establish the uniqueness under a condition of Serrin’s type on the fractional Sobolev spaces.     

Time-Periodic Solutions to the Full Navier–Stokes–Fourier System

We consider the full Navier–Stokes–Fourier system describing the motion of a compressible viscous and heat conducting fluid driven by a time-periodic external force. We show the existence of at least one weak time periodic solution to the problem under the basic hypothesis that the system is allowed to dissipate the thermal energy through the boundary. Such a condition is in fact necessary, as energetically closed fluid systems do not possess non-trivial (changing in time) periodic solutions as a direct consequence of the Second law of thermodynamics.     

Serrin-Type Blowup Criterion for Full Compressible Navier–Stokes System

The authors establish a Serrin-type blowup criterion for the Cauchy problem of the three-dimensional full compressible Navier–Stokes system, which states that a strong or smooth solution exists globally, provided that the velocity satisfies Serrin’s condition and that the temporal integral of the maximum norm of the divergence of the velocity is bounded. In particular, this criterion extends the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier–Stokes equations to the three-dimensional full compressible system and is just the same as that of the barotropic case.     

Asymptotic Stability of Landau Solutions to Navier–Stokes System

It is known that the three-dimensional Navier–Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions that are axisymmetric and homogeneous of degree −1. We show that these solutions are asymptotically stable under any L ²-perturbation.     

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.     

2D Stochastic Navier–Stokes Equations with a Time-Periodic Forcing Term

We study the long time behavior of the solution X(t, s, x) of a 2D-Navier–Stokes equation subjected to a periodic time dependent forcing term. We prove in particular that as , approaches a periodic orbit independently of s and x for any continuous and bounded real function .      

Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

In this paper, we consider the Cauchy problem for a nonlinear parabolic system ${u^\epsilon_t - \Delta u^\epsilon + u^\epsilon \cdot \nabla u^\epsilon + \frac{1}{2}u^\epsilon\, {\rm div}\, u^\epsilon - \frac{1}{\epsilon}\nabla\, {\rm div}\, u^\epsilon = 0}$ in ${\mathbb {R}^3 \times (0,\infty)}$ with initial data in Lebesgue spaces ${L^2(\mathbb {R}^3)}$ or ${L^3(\mathbb {R}^3)}$ . We analyze the convergence of its solutions to a solution of the incompressible Navier?CStokes system as ${\epsilon \to 0}$ .     

The Oberbeck–Boussinesq Approximation as a Singular Limit of the Full Navier–Stokes–Fourier System

We consider the asymptotic limit for the complete Navier–Stokes–Fourier system as both Mach and Froude numbers tend to zero. The limit is investigated in the context of weak variational solutions on an arbitrary large time interval and for the ill-prepared initial data. The convergence to the Oberbeck–Boussinesq system is shown.