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

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.     

On the Existence of Weak Solutions to the Steady Compressible Navier–Stokes Equations in Domains with Conical Outlets

We investigate the steady compressible Navier–Stokes system of equations in the isentropic regime in a domain with several conical outlets and with prescribed pressure drops. Existence of weak solutions is proved and estimate of these solutions with respect to the pressure drops is derived under the hypothesis γ > 3 where γ is the adiabatic constant.     

On Compactness, Domain Dependence and Existence of Steady State Solutions to Compressible Isothermal Navier–Stokes Equations

The steady state system of isothermal Navier–Stokes equations is considered in two dimensional domain including an obstacle. The shape optimisation problem of minimisation of the drag with respect to the admissible shape of the obstacle is defined. The generalized solutions for the Navier–Stokes equations are introduced. The existence of an optimal shape is proved in the class of admissible domains. In general the solutions are not unique for the problem under considerations.     

Existence of Solutions for the Ericksen-Leslie System

We study the general Ericksen-Leslie system, which describes the flow of liquid crystal materials. The dissipation property of the system is established and is used to prove the global existence of weak solutions. We also study the existence of classical solutions and the asymptotic stability of the solutions. (Accepted: January 15, 2000)?Published online September 12, 2000     

Unbounded Branches of Classical Injective Solutions to the Forced Displacement Problem in Nonlinear Elastostatics

We consider classical solutions of the equilibrium equations of nonlinear elastostatics under prescribed boundary displacements and body forces, both of which depend on a loading parameter. We employ recent results on global continuation [HS] that establish general existence of global continua of solutions in nonlinear elastostatics. Under reasonable hypotheses on the stored energy function we prove the existence of an unbounded branch of globally injective solutions to the boundary value problem. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fundamental Solutions of Stokes and Oseen Problem in Two Spatial Dimensions

Fundamental solutions for the linearizations of Stokes and Oseen of the Navier–Stokes time dependent equations in two spatial dimensions are determined. The derivation of these solutions is greatly simplified with the use of a trick known as centering in the probability literature. The relation of these time dependent solutions with their steady counterparts is also established. Authors partially supported by ONR grant N00014-02-1-0116 (RBG) and NSF 0327705 (EAT).     

Propagation of Density-Oscillations in Solutions to the Barotropic Compressible Navier–Stokes System

Considering a bounded sequence of weak solutions to the compressible Navier–Stokes system, we introduce Young measures as in [12] in order to describe a “homogenized system” satisfied in the limit. We then study the Cauchy problem associated to this “homogenized system” when Young measures are convex combinations of Dirac measures.     

On the Long-Time Behaviour of Solutions to the Navier–Stokes–Fourier System with a Time-Dependent Driving Force

In this paper, we give a complete characterization of the asymptotic behaviour of solutions to the Navier–Stokes–Fourier system. We show that either the driving force behaves asymptotically as a gradient of a scalar function, in which case any solution tends to a static state, or the total energy goes to infinity with growing time.     

Higher Integrability of Solutions to Generalized Stokes System Under Perfect Slip Boundary Conditions

We prove an L ^q theory result for generalized Stokes system in a \({\mathcal{C}^{2,1}}\) domain complemented with the perfect slip boundary conditions and under Φ-growth conditions. Since the interior regularity was obtained in Diening and Kaplický (Manu Math 141:336–361, 2013), a regularity up to the boundary is an aim of this paper. In order to get the main result, we use Calderón–Zygmund theory and the method developed in Caffarelli and Peral (Ann Math 130:189–213, 1989). We obtain higher integrability of the first gradient of a solution.     

A Note on the Existence of Solutions to the Oseen System in Lipschitz Domains

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space We show, among other things, that there are two positive constants and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to L^q(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with and u ∈ L^∞(Ω) if a ∈ L^∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W^1-1/q,q(∂Ω), with then ∇u, p ∈ L^q(Ω) and if a ∈ C^0,μ(∂Ω), with μ ∈ [0, α), then also, natural estimates holds.     

Existence of Standing Pulse Solutions to an Inhomogeneous Reaction–Diffusion System

We prove the existence of locally unique, symmetric standing pulse solutions to homogeneous and inhomogeneous versions of a certain reaction–diffusion system. This system models the evolution of photoexcited carrier density and temperature inside the cavity of a semiconductor Fabry–Pérot interferometer. Such pulses represent the fundamental nontrivial mode of pattern formation in this device. Our results follow from a geometric singular perturbation approach, based largely on Fenichel's theorems and the Exchange Lemma.     

Interior Regularity for Solutions to the Modified Navier—Stokes Equations

We discuss interior regularity of solutions to the three-dimensional modified Navier--Stokes equations. In particular, we formulate sufficient conditions that guarantee the local Hölder continuity of the velocity gradient.     

On the Existence of Solutions for Stationary Mean-Field Games with Congestion

Mean-field games (MFGs) are models of large populations of rational agents who seek to optimize an objective function that takes into account their location and the distribution of the remaining agents. Here, we consider stationary MFGs with congestion and prove the existence of stationary solutions. Because moving in congested areas is difficult, agents prefer to move in non-congested areas. As a consequence, the model becomes singular near the zero density. The existence of stationary solutions was previously obtained for MFGs with quadratic Hamiltonians thanks to a very particular identity. Here, we develop robust estimates that give the existence of a solution for general subquadratic Hamiltonians.     

Classical Solutions and the Glassey-Strauss Theorem for the 3D Vlasov-Maxwell System

R. Glassey and W. Strauss have proved in [Arch. Rational Mech. Anal. 92 (1986), 59–90] that C ¹ solutions to the relativistic Vlasov-Maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded. The present paper simplifies their proof.     

Existence of Global Entropy Solutions of a Nonstrictly Hyperbolic System

In this paper we use the theory of compensated compactness coupled with some basic ideas of the kinetic formulation by Lions, Perthame, Souganidis & Tadmor [LPS, LPT] to establish an existence theorem for global entropy solutions of the nonstrictly hyperbolic system (1).     

Existence of Weak Solutions for a Two-dimensional Fluid-rigid Body System

We consider the problem of a rigid body immersed in an inviscid incompressible fluid in two dimensional space. The motion of the fluid is described by the incompressible Euler equations and the motion of the rigid body is governed by the balance of linear and angular momentum. A global weak solution is obtained, without any assumption on the weighted norm of the initial vorticity.     

Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws

The basic existence theory of Kato and Majda enables us to obtain local-in-time classical solutions to generally quasilinear hyperbolic systems in the framework of Sobolev spaces (in x) with higher regularity. However, it remains a challenging open problem whether classical solutions still preserve well-posedness in the case of critical regularity. This paper is concerned with partially dissipative hyperbolic system of balance laws. Under the entropy dissipative assumption, we establish the local well-posedness and blow-up criterion of classical solutions in the framework of Besov spaces with critical regularity with the aid of the standard iteration argument and Friedrichs’ regularization method. Then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin–Lerner spaces (mixed space-time Besov spaces). This fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well-posedness and stability by assuming at all times the Shizuta–Kawashima algebraic condition. As a direct application, the corresponding well-posedness and stability of classical solutions to the compressible Euler equations with damping are also obtained.