Stationary micropolar fluid with boundary data in L

We consider the Dirichlet boundary value problem for the equations of a stationary micropolar fluid in a bounded three-dimensional domain. We show the existence and uniqueness of a distributional solution with boundary values in L².     

The inviscid limit of the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition

In this paper, we study the asymptotic behavior for the incompressible anisotropic Navier–Stokes equations with the non-slip boundary condition in a half space of ${\mathbb{R}^3}$ when the vertical viscosity goes to zero. Firstly, by multi-scale analysis, we formally deduce an asymptotic expansion of the solution to the problem with respect to the vertical viscosity, which shows that the boundary layer appears in the tangential velocity field and satisfies a nonlinear parabolic–elliptic coupled system. Also from the expansion, it is observed that away from the boundary the solution of the anisotropic Navier–Stokes equations formally converges to a solution of a degenerate incompressible Navier–Stokes equation. Secondly, we study the well-posedness of the problems for the boundary layer equations and then rigorously justify the asymptotic expansion by using the energy method. We obtain the convergence results of the vanishing vertical viscosity limit, that is, the solution to the incompressible anisotropic Navier–Stokes equations tends to the solution to degenerate incompressible Navier–Stokes equations away from the boundary, while near the boundary, it tends to the boundary layer profile, in both the energy space and the L ^∞ space.     

Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations

We present a very simple proof of the global existence of a C^∞ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has C^∞ dependence on initial data u₀ in the class of H^s divergence-free vector fields for s > 2.     

On the existence of Lp solutions of the magnetohydrodynamic equations in a bounded domain

In this paper, we show the existence of L^p solutions of the magnetohydrodynamic equations with the non-slip boundary condition and the perfect conducting wall boundary condition. The magnetohydrodynamic equations are a phenomenological model for magnetic fluid.     

On strong solutions of the multi-layer quasi-geostrophic equations of the ocean

In this article, we study the multi-layer quasi-geostrophic equations of the ocean. The existence of strong solutions is proved. We also prove the existence of a maximal attractor in L²(Ω)$L^2\left(\Omega \right)$ and we derive estimates of its Hausdorff and fractal dimensions in terms of the data. Our estimates rely on a new formulation that we introduce for the multi-layer quasi-geostrophic equation of the ocean, which replaces the nonhomogeneous boundary conditions (and the nonlocal constraint) on the stream-function by a simple homogeneous Dirichlet boundary condition. This work improves the results given in [C. Bernier, Existence of attractor for the quasi-geostrophic approximation of the Navier–Stokes equations and estimate of its dimension, Adv. Math. Sci. Appl. 4 (2) (1994) 465–489].     

Solvability,asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

In the paper, we investigate the mixed type transmission problem arising in the model of fluid–solid acoustic interaction when a piezoceramic elastic body (Ω⁺) is embedded in an unbounded fluid domain (Ω^?). The corresponding physical process is described by the boundary‐transmission problem for second‐order partial differential equations. In particular, in the bounded domain Ω⁺, we have a 4×4 dimensional matrix strongly elliptic second‐order partial differential equation, while in the unbounded complement domain Ω^?, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener–Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Highly rotating fluids in rough domains

We consider a rotating fluid in a domain with rough horizontal boundaries. The Rossby number, kinematic viscosity and roughness are supposed of characteristic size ε. We prove a strong convergence theorem on solutions of Navier–Stokes–Coriolis equations, as ε goes to 0, in the well-prepared case. We show in particular that the limit system is a two-dimensional Euler equation with a nonlinear damping term due to boundary layers. We thus give a substantial refinement of the results obtained on flat boundaries with the classical Ekman layers.     

Stationary Solutions of Compressible Navier–Stokes Equations with Slip Boundary Conditions

We consider the Navier–Stokes equations for a compressible, viscous fluid with heat–conduction in a bounded domain of IR² or IR³. Under the assumption that the external force field and the external heat supply are small we prove the existence and local uniqueness of a stationary solution satisfying a slip boundary condition. For the temperature we assume a Dirichlet or an oblique boundary condition.     

Boundary Layers Associated with Incompressible Navier-Stokes Equations: The Noncharacteristic Boundary Case

The goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic, i.e., there is injection and/or suction everywhere at the boundary. Following earlier work on the boundary layer for linearized Navier-Stokes equations in the case where the boundaries are characteristic (no-slip at the boundary and non-permeable), we consider here the case where the boundary is permeable and thus noncharacteristic. The form of the boundary layer and convergence results are derived in two cases: linearized equation and full nonlinear equations. We prove that there exists a boundary layer at the outlet (downwind) of the form e^−Uz/ε where U is the speed of injection/suction at the boundary, z is the distance to the outlet of the channel, and ε is the kinematic viscosity. We improve an earlier result of S. N. Alekseenko (1994, Siberian Math. J.35, No. 2, 209-230) where the convergence in L² of the solutions of the Navier-Stokes equations to that of the Euler equations at vanishing viscosity was established. In the two dimensional case we are able to derive the physically relevant uniform in space (L^∞ norm) estimates of the boundary layer. The uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic Sobolev imbedding. To the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full (nonlinear) Navier-Stokes equations for incompressible fluids.     

Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

We consider the incompressible Euler equations in a (possibly multiply connected) bounded domain Ω of R², for flows with bounded vorticity, for which Yudovich (1963) proved in [29] global existence and uniqueness of the solution. We prove that if the boundary ∂Ω of the domain is C^∞ (respectively Gevrey of order M?1) then the trajectories of the fluid particles are C^∞ (respectively Gevrey of order M+2). Our results also cover the case of “slightly unbounded” vorticities for which Yudovich (1995) extended his analysis in [30]. Moreover if in addition the initial vorticity is Hölder continuous on a part of Ω then this Hölder regularity propagates smoothly along the flow lines. Finally we observe that if the vorticity is constant in a neighborhood of the boundary, the smoothness of the boundary is not necessary for these results to hold.     

Energy decay and global solutions for a damped free boundary fluid–elastic structure interface model with variable coefficients in elasticity

ABSTRACT

We study a free boundary fluid-structure interaction model. In the model, a viscous incompressible fluid interacts with an elastic body via the common boundary. The motion of the fluid is governed by Navier–Stokes equations while the displacement of the elastic structure is described by variable coefficient wave equations. The dissipation is placed on the common boundary between the fluid and the elastic body. Given small initial data, the global existence of the solutions of this system is proved and the exponential decay of solutions is obtained.     

On multiscale homogenization problems in boundary layer theory

This paper is concerned with the homogenization of the equations describing a magnetohydrodynamic boundary layer flow past a flat plate, the flow being subjected to velocities caused by injection and suction. The fluid is assumed incompressible, viscous and electrically conducting with a magnetic field applied transversally to the direction of the flow. The velocities of injection and suction and the applied magnetic field are represented by rapidly oscillating functions according to several scales. We derive the homogenized equations, prove convergence results and establish error estimates in a weighted Sobolev norm and in C ⁰-norm. We also examine the asymptotic behavior of the solutions of the equations governing a boundary layer flow past a rough plate with a locally periodic oscillating structure.     

Existence of global strong solution to the micropolar fluid system in a bounded domain

In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, L^p–L^q type estimates are obtained. By use of the L^p–L^q estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto‐micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd.     

A new algebraic approach to stabilization for boundary control systems of parabolic type

We study the stabilization problem of linear parabolic boundary control systems. While the control system is described by a pair of standard linear differential operators (L,τ), the corresponding semigroup generator generally admits no Riesz basis of eigenvectors. In the sense that very little information on the fractional powers of this generator is needed, our approach has enough generality as a prototype to be used for other types of parabolic systems. We propose in this paper a new algebraic approach to the stabilization, which gives—to the best of the author's knowledge—the simplest framework of the problem. The control system with the scheme of boundary observation/boundary feedback is turned into the differential equations with no boundary input in usual and standard L²-spaces in a readable manner.     

Construction of a Lyapunov functional for 1D-viscous compressible barotropic fluid equations admitting vacua

The Navier-Stokes equations for a compressible barotropic fluid in 1D with zero velocity boundary conditions are considered. We study the case of large initial data in H¹ as well as the mass force such that the stationary density is uniquely determined but admits vacua. Missing uniform lower bound for the density is compensated by a careful modification of the construction procedure for a Lyapunov functional known for the case of solutions which are globally away from zero [I. Straškraba, A.A. Zlotnik, On a decay rate for 1D-viscous compressible barotropic fluid equations, J. Evol. Equ. 2 (2002) 69-96]. An immediate consequence of this construction is a decay rate estimate for this highly singular problem. The results are proved in the Eulerian coordinates for a large class of increasing state functions including p(ρ)=aργ with any γ>0 (a>0 a constant).     

Robust control problems for the multilayer quasi-geostrophic system

In this article, we study some robust control problems associated with the multilayer quasi-geostrophic equations of the ocean and related to data assimilation in oceanography. We consider higher norms (compared to [T. Tachim Medjo, Robust control problems associated with the multilayer quasi-geostrophic equations of the ocean, Appl. Math. Optim. 51(3) (2005) 333–360]) in the definition of the cost functionals. We prove the existence and uniqueness of solutions. The result relies on better a priori estimates on the solutions to the multilayer quasi-geostrophic system obtained using a new formulation that we introduce for the multilayer quasi-geostrophic equation of the ocean. The new formulation replaces the non-homogenous boundary conditions (and the non-local constraint) on the stream-function by a simple homogenous Dirichlet boundary condition.     

Some results for the primitive equations with physical boundary conditions

In this paper, we consider the (simplified) 3-dimensional primitive equations with physical boundary conditions. We show that the equations with constant forcing have a bounded absorbing ball in the H ¹-norm and that a solution to the unforced equations has its H ¹-norm decay to 0. From this, we argue that there exists an invariant measure (on H ¹) for the equations under random kick-forcing.     

Smooth solutions to a class of mixed type Monge–Ampère equations

We prove the existence of C ^∞ local solutions to a class of mixed type Monge–Ampère equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point. Existence of C ^∞ global solutions to a corresponding class of linear mixed type equations is also established. These results are motivated by and may be applied to the problem of prescribed Gaussian curvature for graphs, the isometric embedding problem for 2-dimensional Riemannian manifolds into Euclidean 3-space, and also transonic fluid flow.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

Well-posedness of the Hydrodynamic Model for Semiconductors

This paper concerns the well-posedness of the hydrodynamic model for semiconductor devices, a quasi-linear elliptic–parbolic–hyperbolic system. Boundary conditions for elliptic and parabolic equations are Dirichlet conditions while boundary conditions for the hyperbolic equations are assumed to be well-posed in L² sense. Maximally strictly dissipative boundary conditions for the hyperbolic equations satisfy the assumption of well-posedness in L² sense. The well-posedness of the model under the boundary conditions is demonstrated.