Mixed boundary value problems for steady-state magnetohydrodynamic equations of viscous incompressible fluid

The inhomogeneous boundary value problem for the steady-state magnetohydrodynamic equations of viscous incompressible fluid under the Dirichlet conditions for the velocity and mixed boundary conditions for the electromagnetic field is considered. Sufficient conditions for the data that ensure the global solvability of this problem and the local uniqueness of its solution are found.     

Solvability of an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid

We study an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid corresponding to the case in which the tangential component of the magnetic field is specified on the boundary and the Dirichlet condition is posed for the velocity. We derive sufficient conditions on the input data for the global solvability of the problem and the local uniqueness of the solution.     

Solution of the boundary value problem for the Navier-Stokes equation for the flow of a gaseous medium past a heated spheroid

We find the solution of the Navier-Stokes equation linearized with respect to the velocity with regard of a power-law dependence of the molecular transport coefficients (viscosity and heat conductivity) and the gaseous medium density on the temperature. The uniqueness of the solution is proved.     

Leray's problem for a viscous incompressible micropolar fluid

This work concerns the steady motion of a viscous incompressible micropolar fluid in unbounded domains having cylindrical outlets to infinity. We prove the existence of a solution that approaches prescribed parallel solutions along the outlets of the domain. We also study the uniqueness, the regularity and the asymptotic behavior of the solution.     

Analysis of boundary conditions for approximate solution of problems of viscous incompressible flow past a body

Translated from Aktual'nye Voprosy Prikladnoi Matematiki, pp. 49–54, 1989.     

On the motion of rigid bodies in a viscous incompressible fluid

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.AMS Subject Classification: 49Q10m, 35P15, 49J20.     

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

Free boundary problem for a two-layer inviscid incompressible fluid

The existence of a local (in time) classical solution of a free boundary problem for a two-layer inviscid incompressible fluid is shown. The method of successive approximations and the novel approach to Lagrangian coordinates of Solonnikov are used.     

Solvability of a nonstationary problem of a rigid body motion in the flow of a viscous incompressible fluid in a pipe of arbitrary cross-section

The existence of a generalized weak solution is proved for the nonstationary problem of motion of a rigid body in the flow of a viscous incompressible fluid filling a cylindrical pipe of arbitrary cross-section. The fluid flow conforms to the Navier–Stokes equations and tends to the Poiseuille flow at infinity. The body moves in accordance with the laws of classical mechanics under the influence of the surrounding fluid and the gravity force directed along the cylinder. Collisions of the body with the boundary of the flow domain are not admitted and, by this reason, the problem is considered until the body approaches the boundary.     

High-order approximations for an incompressible viscous flow on a rough boundary

In this paper, we propose approximations of fluid flow that could be used for obtaining wall laws of higher order. We consider the two-dimensional laminar fluid flow, modeled by the incompressible Stokes system in a straight channel with a rough side. The roughness is periodic and the ratio of the amplitude of the rough part and the size of the flow domain is denoted by ?, being a small number. We impose periodic boundary conditions on the flow. We generalize the boundary layers needed for the construction of flow approximations of higher order with respect to ?. The existence of the layers and their features are discussed. Finally we give the error estimates for the approximations and establish an explicit wall law.     

On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω=Ω₀×(0,L)∈R³. We show existence of a solution , p>3, where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of . The term u⋅∇w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (vn,ρn) that is bounded in and satisfies the Cauchy condition in a larger space L_∞(0,L;L₂(Ω₀)) what enables us to deduce that the weak limit of a subsequence of (vn,ρn) is in fact a strong solution to our problem.     

Solvability of a problem on the plane motion of a viscous incompressible fluid with a free noncompact boundary

One considers the problem of the plane motion of a viscous incompressible fluid which fills partially a container V, bounded by the straight line ₁ = {x:x ₂ = 0} and the contour (V₁), consisting of two semilines ⁽¹⁾ = {x:x ₁<–1,x ₂ = h₀} ⁽²⁾ = {x:x ₁ = 0,x ₂h₀+1} joined by a smooth curvel ⁽³⁾. One assumes that the motion is due to a nonzero flow and by the motion of the lower wall ₁ with a constant velocity R0. The unique solvability of this problem is proved for small R and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 174–179, 1981.In conclusion, the author expresses his deep gratitude to V. A. Solonnikov for his guidance.     

Effect of a viscous fluid flow past a spherical gas bubble on the growth of its radius

A nearly spherical gas bubble expands adiabatically in a viscous incompressible fluid flowing past it. The Rayleigh-Plesset formula for the growth of the bubble radius is modified due to the flow of the viscous fluid.     

On the slow motion of a self-propelled rigid body in a viscous incompressible fluid

In this paper we study the Stokes approximation of the self-propelled motion of a rigid body in a viscous liquid that fills all the three-dimensional space exterior to the body. We prove the existence and uniqueness of strong solution to the coupled systems of equations describing the motion of the system body-liquid, for any time and any regular distribution of velocity on the boundary of the body. For the corresponding stationary problem we derive L^p-estimates for the solution in terms of the data. Finally, we prove that every steady solution is attainable as the limit, when t→∞, of an unsteady self-propelled solution which starts from rest.     

On the optimal control of viscous incompressible fluid flow

The framework of the Navier-Stokes (N-S) equations is used to study flow past an arbitrary body on whose surface the tangential or normal velocity is under control. The necessary conditions are obtained for the minimum rate of energy dissipation. Exact analytical solutions of the corresponding problems are found for the case of flow past an ellipsoid in the Stokes approximation.     

On the existence of free surface problem for viscous incompressible flow

In this paper we are concerned with the flow of a viscous, incompressible fluid in a bounded, three-dimensional region Ω with free surface boundary conditions. Using a method introduced by the author, that consider a two-fluid system in which the atmosphere or the vacuum is considered as a second fluid, separated from the first one by a free interface Γ(t), we prove existence of a kind of weak solution that we call quasi-weak solution.

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Sunto In questo lavoro studiamo il moto di un fluido viscoso e incomprimibile in una regione limitata tridimensionale Ω, con condizioni al contorno di superficie libera. Utilizzando un metodo, dovuto all’autore, che consiste nel considerare l’atmosfera o il vuoto come un secondo fluido, separato dal primo da un’interfaccia mobile Γ(t), dimostriamo l’esistenza di una sorta di soluzione debole, denominata soluzione quasi-debole.

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Work supported by Progetto Murst n. 9801262841.     

On the initial‐boundary value problem of the incompressible viscoelastic fluid system

In this paper, we shall establish the local well‐posedness of the initial‐boundary value problem of the viscoelastic fluid system of the Oldroyd model. We shall also prove that the local solutions can be extended globally and that the global solutions decay exponentially fast as time goes to infinity provided that the initial data are sufficiently close to the equilibrium state. © 2007 Wiley Periodicals, Inc.