Volume 35, Issue 4

We consider the nonstationary nonlinear Navier–Stokes equations describing the motion of a viscous incompressible fluid flow for 0<t≤T${0 &lt; t \le T}$ in a bounded domain Ω⊆ℝ3${\Omega \subseteq \mathbb {R}^3}$ with sufficiently smooth boundary ∂Ω${\partial \Omega }$. We use a particle method in connection with a time delay to approximate the nonlinear convective term by a single central Lagrangian difference quotient constructed from autonomous systems of ordinary differential equations. We show that the resulting approximate Navier–Stokes system has a uniquely determined global solution satisfying the energy equation and having a high degree of regularity uniformly in time. Moreover, we prove that the sequence of approximate solutions has an accumulation point satisfying the Navier–Stokes equations in a weak sense and the energy inequality.

In this paper we study the compressible Navier–Stokes limit of binary mixture of gas particles in which a species is dense and the other is sparse. Their collisions are decided by Grad's hard potentials. When the Knudsen number of dense species of the Boltzmann system goes to zero, we show that the hydrodynamic variables satisfy compressible Navier–Stokes type equations. It turns out that the macro fluid variables corresponding to the dense species satisfy the standard compressible Navier–Stokes equations. But the fluid equations for sparse species contain influence terms of dense species. Like single species gas, we employed Enskog–Chapman and moment methods up to the first order.

In this paper we prove the asymptotic stability of small global solutions in the weak L 2 space for the two-dimensional Navier–Stokes equations under a natural functional setting for the class of perturbations. Our result is a global one, that is, we do not impose any smallness on the perturbations.

In the present paper we prove that a weak Leray solution to the Navier–Stokes equations in ℝ3×(0,T]$\mathbb {R}^3\times (0,T]$ is regular provided the gradient of one component belongs to the Prodi–Serrin class ∇u3∈L4(0,T;L2)$\nabla u^3 \in L^4(0,T; L^2)$.

We show existence and uniqueness of steady-state solutions to the equations of generalized Newtonian liquids of shear-thinning type in an unbounded region that includes semi-infinite cylinders of constant cross-section (pipeline system). This result is established under the assumption that the datum (flow-rate) is sufficiently small. The main feature of our approach is the proof of a “global compactness” property of the sequence of approximating solutions.

In this paper, we consider a quasilinear parabolic system of equations describing coupled bulk and interface diffusion, including mixed boundary conditions. The setting naturally includes non-smooth domains Ω. We show local well-posedness using maximal L s -regularity in dual Sobolev spaces of type W-1,q(Ω)$W^{-1,q}(\Omega )$ for the associated abstract Cauchy problem.

We present the study of systems of equations governing a steady flow of polyatomic, heat-conducting reactive gas mixture. It is shown that the corresponding system of PDEs admits a weak solution and renormalized solution to the continuity equation, provided the adiabatic exponent for the mixture γ is greater than 53${\frac{5}{3}}$.

In this paper we show how the stability of Prandtl boundary layers is linked to the stability of shear flows in the incompressible Navier–Stokes equations. We then recall classical physical instability results, and give a short educational presentation of the construction of unstable modes for Orr–Sommerfeld equations. We end the paper with a conjecture concerning the validity of Prandtl boundary layer asymptotic expansions.