H. Beirão da Veiga

On the Existence of Strong Solutions to a Coupled Fluid-Structure Evolution Problem

Abstract We consider here a model of fluid-structure evolution problem which, in particular, has been largely studied from the numerical point of view. We prove the existence of a strong solution to this problem.

Navier–Stokes Equations with Shear Thinning Viscosity. Regularity up to the Boundary

Abstract.In this article we prove some sharp regularity results for the stationary and the evolution Navier–Stokes equations with shear dependent viscosity, see (1.1), under the no-slip boundary condition(1.4). We are interested in regularity results for the second order derivatives of the velocity and for the first order derivatives of the pressure up to the boundary, in dimension n ≥ 3. In reference [4] we consider the stationary problem in the half space

A sufficient condition on the pressure for the regularity of weak solutions to the Navier-Stokes equations

{\mathbb{R}}_+^n

The stability of one dimensional stationary flows of compressible viscous fluids

under slip and no-slip boundary conditions. Here, by working in a simpler context, we concentrate on the basic ideas of proofs. We consider a cubic domain and impose our boundary condition (1.4) only on two opposite faces. On the other faces we assume periodicity, as a device to avoid unessential technical difficulties. This choice is made so that we work in a bounded domain Ω and, at the same time, with a flat boundary. In the last section we provide the extension of the results from the stationary to the evolution problem.

Concerning the Existence of Classical Solutions to the Stokes System. On the Minimal Assumptions Problem

We improve regularity criteria for weak solutions to the Navier-Stokes equations stated in references [1], [3] and [12], by using in the proof given in [3], a new idea introduced by H. O. Bae and H. J. Choe in [1]. This idea allows us, in one of the main hypothesis (see eq. (1.7)), to replace the velocity u by its projection \( \bar u \) into an arbitrary hyperplane of \( {\Bbb R}^n \); see Theorem A. For simplicity, we state our results for space dimension \( n \le 4 \), since if \( n \ge 5 \) the proofs become more technical and additional hypotheses are needed. However, for the interested reader, we will present the formal calculations for arbitrary dimension n.

On the Existence Theorem for the Barotronic Motion of a Compressible Inviscid Fluid in the Half-Space (*).

It is well known that a weak solution (v, p) to the Navier-Stokes equations is regular if v satisfies some suitable extra conditions (see (1.2), (1.3)). However, with the exception of the recent papers [BV4], [BV5] (see also [K], [Be]) not so much attention has been payed to “alternative natural assumptions” that p may fulfill, in order that (v, p) be regular. By “alternative natural assumptions”, we mean assumptions that formally follow from the Poisson equation relating pressure and velocity (see (1.4)). The objective of this paper is to prove that (v, p) is regular if

Attracting properties for one dimensional flows of a general barotropic viscous fluid. Periodic flows

|p|/(1 + |v|)

On Some Boundary Value Problems for Flows with Shear Dependent Viscosity

obeys some conditions that are in formal agreement with this relation.