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 $ |p|/(1 + |v|) $ obeys some conditions that are in formal agreement with this relation.

We consider the Euler equations of barotropic inviscid compressible fluids in a bounded domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In this paper we discuss, for the boundary case, the different kinds of convergence under various assumptions on the data, in particular the weak convergence in the case of uniformly bounded initial data and the strong convergence in the norm of the data space.

The Navier—Stokes equations for incompressible fluids are coupled to models of reduced complexity, such as Oseen and Stokes, and the corresponding transmission conditions are investigated. A mathematical analysis of the corresponding problems is carried out. Numerical results obtained by finite elements and spectral elements are shown on several flow fields of physical interest.

In this paper the classical method to prove a removable singularity theorem for harmonic functions near an isolated singular point is extended to solutions to the stationary Stokes and Navier—Stokes system. Finding series expansion of solutions in terms of homogeneous harmonic polynomials, we establish some known results and new theorems concerning the behavior of solutions near an isolated singular point. In particular, we prove that if (u, p) is a solution to the Navier—Stokes system in $ B_R \setminus \{0\} $ , $ n \geq 3 $ and $ |u(x)| = o\,(|x|^{-(n - 1)/2}) $ as $ |x| \to 0 $ or $ u \in L^{2n/(n - 1)}(B_R) $ , then (u, p) is a distribution solution and if in addition, $ u \in L^{\beta}(B_R) $ for some $ \beta > n $ then ( u, p) is smooth in B R .

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.

We consider a stationary problem for the Navier—Stokes equations in a domain $ \Omega\subset R^2 $ with a finite number of "outlets" to infinity in the form of infinite sectors. In addition to the standard adherence boundary conditions, we prescribe total fluxes of velocity vector field in each "outlet", subject to the necessary condition that the sum of all fluxes equals zero. Under certain restrictions on the aperture angles of the "outlets", which seem close to being necessary, we prove that for small fluxes this problem has a solution which behaves at infinity like the Jeffery—Hamel flow with the same flux, and we prove that this solution is unique in the class of solutions satisfying the energy inequality (4.4). We also study the problem with another type of additional condition at infinity, that which involves limiting values of the pressure at infinity in the outlets. Finally, we present a simplified construction of a small Jeffery—Hamel solution with a given flux based on the contraction mapping principle.

We consider physically reasonable solutions of the stationary Navier—Stokes system in a three-dimensional exterior domains with zero velocity at infinity. We show that when these solutions are asymptotically expanded near infinity, the leading term cannot be the product of a non-zero vector with the Stokes fundamental solution. This result should be contrasted with the case when the velocity at infinity is not zero. Then, as is well known, such an expansion is possible, with the leading term being the product of a suitable constant vector with the fundamental Oseen solution.

We show that if v is an axially symmetric suitable weak solution to the Navier—Stokes equations (in the sense of L. Caffarelli, R. Kohn & L. Nirenberg — see [2]) such that either $ v_{\rho} $ (the radial component of v) or $ v_{\theta} $ (the tangential component of v) has a higher regularity than is the regularity following from the definition of a weak solution in a sub-domain D of the time-space cylinder Q T then all components of v are regular in D.

An example is presented of a class of periodic, two-dimensional, inviscid fluid flows where the stability spectrum contains both discrete unstable eigenvalues and an unstable essential spectrum. The method of averaging is used to demonstrate the existence of unstable eigenvalues. For such flows spectral instability implies nonlinear instability.

We study the motion of a rigid body of arbitrary shape immersed in a viscous incompressible fluid in a bounded, three-dimensional domain. The motion of the rigid body is caused by the action of given forces exerted on the fluid and on the rigid body. For this problem, we prove the global existence of weak solutions.

The equations governing the motion of incompressible viscoelastic fluids of Rivlin—Ericksen and Oldroyd type are investigated in domains with cylindrical and paraboloidal outlets to infinity. For sufficiently small fluxes, prescribed in each outlet, existence and uniqueness of solutions are proven in weighted Hölder spaces. In domains with paraboloidal outlets the solution is obtained as a perturbation of the corresponding Navier—Stokes solution and in domains with cylindrical outlets as a perturbation of a flux carrier, constructed by joining together the exact solutions found in each outlet. These exact solutions are shown to be either rectilinear flows of Poiseuille type or flows composed of a rectilinear and of a transverse secondary component.

In the present paper, we shall consider a nonlinear thermoconvection problem consisting of a coupled system of nonlinear partial differential equations due to temperature dependent coefficients. We prove that weak solutions exist in appropriate Sobolev spaces under mild hypothesis on the regularity of the data. This result is established through a fixed point theorem for multivalued functions, which requires a detailed analysis of the continuous dependence of auxiliary problems, including the associated Lagrange multipliers of the generalized Navier—Stokes system.

We consider the three-dimensional steady flow of certain classes of viscoelastic fluids in exterior domains with non-zero velocity prescribed at infinity. We show that the solution behaves near infinity similarly as the fundamental solution to the Oseen problem.

In this paper we study the strong solvability of the Navier—Stokes equations for rough initial data. We prove that there exists essentially only one maximal strong solution and that various concepts of generalized solutions coincide. We also apply our results to Leray—Hopf weak solutions to get improvements over some known uniqueness and smoothness theorems. We deal with rather general domains including, in particular, those having compact boundaries.

We consider a regularized one-dimensional hydrodynamical model of nuclear slab, with a Van-der-Waals type pressure law, for which we identify asymptotically stable stationary states, and we prove global existence, for small data.¶We can also describe the asymptotic behaviour of the system for large time, provided that the initial density is restricted to a pure phase region.