Abstract. The asymptotic structure of laminar modulated travelling waves in two-dimensional high-Reynolds-number plane Poiseuille flow is investigated on the upper-energy branch. A finite set of independent slowly varying parameters are identified which parameterize the solution of the Navier–Stokes equations in this subset of the phase space. Our parameterization of the weakly stable modes des...

We study the asymptotic behavior of solutions of the evolution Stokes equation in a thin three-dimensional domain bounded by two moving surfaces in the limit as the distance between the surfaces approaches zero. Using only a priori estimates and compactness it is rigorously verified that the limit velocity field and pressure are governed by the time-dependent Reynolds equation.

We consider the Navier–Stokes equations for compressible isothermal flow in the steady two dimensional case and show the existence of a weak solution in the case of periodic and of mixed boundary conditions.

In this article, a non linear family of spaces, based on the energy dissipation, is introduced. This family bridges an energy space (containing weak solutions to Navier-Stokes equation) to a critical space (invariant through the canonical scaling of the Navier-Stokes equation). This family is used to get uniform estimates on higher derivatives to solutions to the 3D Navier-Stokes equations. Tho...

For the 3D Navier–Stokes problem on the whole space, we study existence, regularity and stability of time-periodic solutions in Lebesgue, Lorentz or Sobolev spaces, when the periodic forcing belongs to critical classes of forces.

Estimates for the three α-models known as the LANS-α, Leray-α and Bardina models are found in terms a Reynolds number associated with a Navier-Stokes velocity field. They are tabulated for comparative purposes and show clearly that all estimates for the Leray-α model are smaller than those for the LANS-α and Bardina models.

We consider the open problem of regularity for L3,∞-solutions to the Navier-Stokes equations. We show that the problem can be reduced to a backward uniqueness problem for the heat operator with lower order terms. 1991 Mathematical subject classification (Amer. Math. Soc.): 35K, 76D.

In this paper we prove the nonexistence of global weak solutions to the compressible Navier-Stokes equations for the isentropic gas in R N , N ≥ 3, where the pressure law given by p(ρ) = aρ γ , a > 0, 1 < γ ≤

We study spatial analyticity properties of solutions of the Navier-Stokes equation and obtain new growth rate estimates for the analyticity radius. We also study stability properties of strong global solutions of the Navier-Stokes equation with data in H, r ≥ 1/2 and prove a stability result for the analyticity radius.

This paper concerns the two-dimensional NavierStokes equations in a Lipschitz domain Ω with nonhomogeneous boundary condition u = φ on ∂Ω. Assuming φ ∈ L∞(∂Ω), we establish the existence of the universal attractor, and show that its dimension is bounded by c1G + c2Re, where G is the Grashof number and Re the Reynolds number.