We consider perturbations to the rest state of heat conductive compressible fluid in a three-dimensional exterior domain. In the regularity class $ {\cal V}_s $ , we show stability of L 2 -norms of these perturbations, while in the more restrictive class $ {\cal V}_d $ we show convergence to zero of these perturbations along a time sequence converging to infinity.

We consider the zero-velocity stationary problem of the Navier—Stokes equations of compressible isentropic flow describing the distribution of the density $ \varrho $ of a fluid in a spatial domain $ \Omega \subset {\rm R}^N $ driven by a time-independent potential external force $ \vec f = \triangledown F $ . We study the structure of the set of all solutions to the stationary problem having a prescribed mass m > 0 and a prescribed energy. Cardinality of the solution set depends on m and it is either continuum or at most two. Conditions on m for distinguishing these cases have been found. Uniqueness for the stationary system is also studied.

We consider local mesh adaptation for stationary flow control problems. The state equation is given by the incompressible Navier-Stokes equations with Neumann boundary control. As an example, the functional to be minimized is the drag coefficient of an immersed body. We use finite elements on locally refined meshes to discretize the first order necessary conditions based on the Lagrangian. An a posteriori error estimate is derived which is directly related to the control problem. It is used to successively enrich the finite element space until the computed solution satisfies prescribed accuracy requirements.

We prove the existence of globally defined weak solutions to the Navier—Stokes equations of compressible isentropic flows in three space dimensions on condition that the adiabatic constant satisfies $ \gamma > 3/2 $ .

We prove an optimal relationship between the regularity of a function and the asymptotic behavior of its Fourier transform. As an application of this result we show L p -estimates for the Stokes semigroup in $ \Bbb R^n $ and $ {\Bbb R}_+^n $ when $ 1\leqq p\leqq \infty $ .

For 2D Navier—Stokes equations defined in a bounded domain $ \Omega $ we study stabilization of solution near a given steady-state flow $ \hat v(x) $ by means of feedback control defined on a part $ \Gamma $ of boundary $ \partial\Omega $ . New mathematical formalization of feedback notion is proposed. With its help for a prescribed number $ \sigma > 0 $ and for an initial condition v 0(x) placed in a small neighbourhood of $ \hat v(x) $ a control u(t,x'), $ x' \in \Gamma $ , is constructed such that solution v(t,x) of obtained boundary value problem for 2D Navier—Stokes equations satisfies the inequality: $ \|v(t,\cdot)-\hat v\|_{H^1}\leqslant ce^{-\sigma t}\quad {\rm for}\; t \geqslant 0 $ . To prove this result we firstly obtain analogous result on stabilization for 2D Oseen equations.

A global-in-time unique smooth solution is constructed for the Cauchy problem of the Navier—Stokes equations in the plane when initial velocity field is merely bounded not necessary square-integrable. The proof is based on a uniform bound for the vorticity which is only valid for planar flows. The uniform bound for the vorticity yields a coarse globally-in-time a priori estimate for the maximum norm of the velocity which is enough to extend a local solution. A global existence of solution for a q-th integrable initial velocity field is also established when $ q > 2 $ .

We investigate the steady motion of a liquid in a lake, modeled as a thin domain. We assume the motion is governed by Navier—Stokes equations, while a Robin-type traction condition, and a friction condition is prescribed at the surface and at the bottom, respectively. We also take into account Coriolis forces. We derive an asymptotic model as the aspect ratio $ \delta $ = depth/width of the domain goes to 0. When the Reynolds number is not too large, this is mathematically justified and the three-dimensional limit velocity is given in terms of wind, bathymetry, depth and of a two-dimensional potential. Numerical simulation is carried out and the influence of traction condition reading is experienced.

Some conceptual ambiguities in the derivation of the equations of capillarity on the basis of the principle of virtual work are addressed, and hypotheses are proposed toward obtaining a physically correct characterization in general circumstances. It is shown that under the hypotheses, the classical equations of capillarity for an interface of an incompressible fluid with a fluid of negligible density can be obtained on the basis of global phenomenological reasoning, without recourse to consideration of intermolecular attractions. More generally, the procedure is applied to derive the specific equations arising from a compressible fluid configuration with idealized pressure-density relationship in a capillary tube, and a general necessary condition for existence of a solution is established. It is shown that for symmetric domains, the condition is also sufficient for existence of a unique symmetric solution.

Let $ \cal B $ ; be a homogeneous body of revolution around an axis a, with fore-and-aft symmetry. Typical examples are bodies of constant density having the shape of cylinders of circular cross-section, of prolate and oblate spheroids, etc. In this paper we prove that, provided a certain geometric condition is satisfied, the only possible orientations that $ \cal B $ ; can eventually achieve when dropped in a Navier-Stokes fluid under the action of the acceleration of gravity g and at a small and nonzero Reynolds number, is with a either parallel or perpendicular to g. This result is obtained by a rigorous calculation of the torque exerted by the fluid on the body. We also show that the above geometric condition is certainly satisfied if $ \cal B $ ; is a prolate spheroid. Moreover, in this case, we prove, by a "quasi-steady" argument, that, at first order in λ, the configuration with a perpendicular to g is stable to small disorientation, while the other is unstable, in accordance with experiments.

This paper is concerned with the Navier-Stokes flows in the homogeneous spaces of degree -1, the critical homogeneous spaces in the study of the existence of regular solutions for the Navier-Stokes equations by means of linearization. In order to narrow the gap for the existence of small regular solutions in $ \dot B^{-1}_{\infty,\infty}(R^n)^n $ , the biggest critical homogeneous space among those embedded in the space of tempered distributions, we study small solutions in the homogeneous Besov space $ \dot B^{-1+n/p}_{p,\infty}(R^n)^n $ and a homogeneous space defined by $ \hat M_n(R^n)^n $ , which contains the Morrey-type space of measures $ \tilde M_n(R^n)^n $ appeared in Giga and Miyakawa [20]. The earlier investigations on the existence of small regular solutions in homogeneous Morrey spaces, Morrey-type spaces of finite measures, and homogeneous Besov spaces are strengthened. These results also imply the existence of small forward self-similar solutions to the Navier-Stokes equations. Finally, we show alternatively the uniqueness of solutions to the Navier-Stokes equations in the critical homogeneous space $ C([0,\infty);L_n(R^n)^n) $ by applying Giga-Sohr's $ L_p(L_q) $ estimates on the Stokes problem.

In a previous paper, the present authors studied the asymptotic behaviour of solutions to steady compressible Navier-Stokes equations in barotropic and isothermal regime with sufficiently small external data, in particular, in the whole plane. Here, the same problem is investigated in a two dimensional exterior domain with the prescribed velocity at infinity $ v_{\infty}\ne 0 $ . Similar results as in Dutto and Novotny [DuNo] are found; in particular, it is proved that there exists a unique solution which possesses the similar pointwise decay and the wake structure as the fundamental Oseen tensor.

The boundary-value problems for the stationary Boussinesq heat transfer equations with general non-standard boundary conditions for the velocity and mixed boundary conditions for the temperature are considered. The local and global existence theorems are proved. The precise a priori estimates for the solution are derived.

In this paper we prove a new energy inequality for weak solutions of Leray—Hopf type for the three-dimensional Navier—Stokes equations. It implies a result of partial regularity.

An example of stratified shear flow is presented in which an explicit construction is given for unstable eigenvalues with smooth eigenfunctions for the Taylor—Goldstein equation. It is proved for any stratified, plane parallel shear flow that the unstable spectrum of the linear operator is purely discrete. A general theorem is then invoked to prove that the specific example is nonlinearly unstable. A sufficient condition for nonlinear stability for stratified shear flow is discussed.

The time-dependent Navier—Stokes problem on an interior or exterior smooth domain, with nonhomogeneous Dirichlet boundary condition, is treated in anisotropic L p Sobolev spaces (1 {\frac 1p} - 1 $ ; the present work extends the solvability to spaces with $ s > {\frac 1p} - 2 $ for zero initial data ( $ s > - 2 $ if f = 0), $ s > {\frac 2p} - 2 $ for nonzero initial data, with s,p subject to other conditions stemming from the nonlinearity.

The paper is concerned with the modelling of viscous incompressible flow in an unbounded exterior domain with the aid of the coupling of the nonlinear Navier—Stokes equations considered in a bounded domain with the linear Oseen system in an exterior domain. These systems are coupled on an artificial interface via suitable transmission conditions. The present paper is a continuation of the work [8], where the coupling of the Navier—Stokes problem with the Stokes problem is treated. However, the coupling "Navier—Stokes — Oseen" is physically more relevant. We give the formulation of this coupled problem and prove the existence of its weak solution for large data.