Free-Boundary Regularity for a Problem Arising in Superconductivity

This paper concerns regularity properties of the mean-field theory of superconductivity. The problem is reminiscent of the one studied earlier by L.A. Caffarelli, L. Karp and H. Shahgholian in connection with potential theory. The difficulty introduced in this paper is the existence of several patches, where on each patch the solution to the problem may have different constant values. However, using a refined analysis, we reduce the problem to the one-patch case, at least locally near regular free boundary points. Using a monotonicity formula, due to Georg S. Weiss, we characterize global solutions of a related equation. Hence earlier regularity results apply and we conclude the C ¹ regularity of the free boundary.     

Stability of Solutions of a Degenerate Linear System of Differential Equations

We introduce the concept of stability of solutions of a system of linear differential equations with an identically degenerate matrix as the coefficient of the derivative. We find necessary and sufficient conditions for the stability of such systems. We generalize the Floquet–Lyapunov theory to systems of this type with periodic coefficients.     

Global Behavior of Compressible Navier-Stokes Equations with a Degenerate Viscosity Coefficient

In this paper, we study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity. Precisely, the viscosity coefficient μ is proportional to ρ ^θ with , where ρ is the density, and γ > 1 is the physical constant of polytropic gas. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give the uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we estimate the stabilization rate in L ^∞ norm, (weighted) L ² norm and weighted H ¹ norm of the solution as time tends to infinity.     

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.     

On the Influence of Viscosity on Riemann Solutions

We show how the existence and uniqueness of Riemann solutions are affected by the precise form of viscosity which is used to select shock waves admitting a viscous profile. We study a complete list of codimension-1 bifurcations that depend on viscosity and distinguish between Lax shock waves with and without a profile. These bifurcations are the saddle–saddle heteroclinic bifurcation, the homoclinic bifurcation, and the nonhyperbolic periodic orbit bifurcation. We prove that these influence the existence and uniqueness of Riemann solutions and affect the number and type of waves comprising a Riemann solution. We present generic situations in which viscous Riemann solutions differ from Lax solutions.     

Non-existence of Entire Solutions of Degenerate Elliptic Inequalities with Weights

Non-existence results for non-negative distribution entire solutions of singular quasilinear elliptic differential inequalities with weights are established. Such inequalities include the capillarity equation with varying gravitational field h, as well as the general p-Poisson equation of radiative cooling with varying heat conduction coefficient g and varying radiation coefficient h. Since we deal with inequalities and positive weights, it is not restrictive to assume h radially symmetric. Theorem 1 extends in several directions previous results and says that solely entire large solutions can exist, while Theorem 2 shows that in the p-Laplacian case positive entire solutions cannot exist. The results are based on some qualitative properties of independent interest. An erratum to this article can be found at     

Convergence to Separate Variables Solutions for a Degenerate Parabolic Equation with Gradient Source

The large time behaviour of nonnegative solutions to a quasilinear degenerate diffusion equation with a source term depending solely on the gradient is investigated. After a suitable rescaling of time, convergence to a unique profile is shown for global solutions. The proof relies on the half-relaxed limits technique within the theory of viscosity solutions and on the construction of suitable supersolutions and barrier functions to obtain optimal temporal decay rates and boundary estimates. Blowup of weak solutions is also studied.     

Zero Viscosity Limit for Analytic Solutions of the Primitive Equations

The aim of this paper is to prove that the solutions of the primitive equations converge, in the zero viscosity limit, to the solutions of the hydrostatic Euler equations. We construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic Euler equation and boundary layer correctors as the first order term, and an error that we show to be \({O(\sqrt{\nu})}\). The main assumption is spatial analyticity of the initial datum.     

Long-Time Behavior of a Coupled Heat-Wave System Arising in Fluid-Structure Interaction

This paper is devoted to the analysis of the long-time behavior of a coupled wave-heat system in which a wave and a heat equation evolve in two bounded domains, with natural transmission conditions at a common interface. These conditions couple, in particular, the heat unknown with the velocity of the wave solution. This model may be viewed as a simplified version of linearized models which arise in fluid-structure interaction. First, we show the strong asymptotic stability of solutions to this system. Then, based on the construction of ray-like solutions by means of geometric optics expansions and a careful analysis of the transfer of energy at the interface, we show the lack of uniform decay in general domains. Further, we obtain a polynomial decay result for smooth solutions of the system under a suitable geometric assumption which guarantees that the heat domain envelopes the wave domain. Finally, in the absence of geometric conditions we show a logarithmic decay result for the same system but with simplified transmission conditions at the interface. We also analyze the difficulty there is to extend this result to the more natural transmission conditions.     

On Solutions of Some Non-Linear Differential Equations Arising in Newtonian and Non-Newtonian Fluids

Some steady as well as unsteady solutions of the equations of motion for an incompressible Newtonian and non-Newtonian (second-grade) fluids are obtained by applying different methods including the Lie symmetry group method. The flows considered are axially symmetric with the swirling motion, and the governing equations for second-grade fluid flow have been modeled. Expressions for streamlines, velocity and vorticity components are constructed explicitly in each case. Exact analytical solutions in second-grade fluid are obtained and compared with the corresponding viscous solutions.     

Viscosity Solutions of Dynamic-Programming Equations for the Optimal Control of the Two-Dimensional Navier-Stokes Equations

In this paper we study the infinite-dimensional Hamilton-Jacobi equation associated with the optimal feedback control of viscous hydrodynamics. We resolve the global unique solvability problem of this equation by showing that the value function is the unique viscosity solution.     

Eigenvalues of Self-Similar Solutions of the Dafermos Regularization of a System of Conservation Laws via Geometric Singular Perturbation Theory

The Dafermos regularization of a system of n conservation laws in one space dimension admits smooth self-similar solutions of the form u=u(X/T). In particular, there are such solutions near a Riemann solution consisting of n possibly large Lax shocks. In Lin and Schecter (2004, SIAM. J. Math. Anal. 35, 884–921), eigenvalues and eigenfunctions of the linearized Dafermos operator at such a solution were studied using asymptotic expansions. Here we show that the asymptotic expansions correspond to true eigenvalue–eigenfunction pairs. The proofs use geometric singular perturbation theory, in particular an extension of the Exchange Lemma.     

Traveling Wave Solutions in a Generalized Theory for Macroscopic Capillarity

One-dimensional traveling wave solutions for imbibition processes into a homogeneous porous medium are found within a recent generalized theory of macroscopic capillarity. The generalized theory is based on the hydrodynamic differences between percolating and nonpercolating fluid parts. The traveling wave solutions are obtained using a dynamical systems approach. An exhaustive study of all smooth traveling wave solutions for primary and secondary imbibition processes is reported here. It is made possible by introducing two novel methods of reduced graphical representation. In the first method the integration constant of the dynamical system is related graphically to the boundary data and the wave velocity. In the second representation the wave velocity is plotted as a function of the boundary data. Each of these two graphical representations provides an exhaustive overview over all one-dimensional and smooth solutions of traveling wave type, that can arise in primary and secondary imbibition. Analogous representations are possible for other systems, solution classes, and processes.     

Asymptotic Stability of Riemann Solutions for a Class of Multidimensional Systems of Conservation Laws with Viscosity

We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. In particular, we obtain the uniqueness of the self-similar L^∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in L_loc¹ of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in L_loc¹(ℝⁿ), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem.     

On the Existence of a Smooth Bounded Semiinvariant Manifold for a Degenerate Nonlinear System of Difference Equations in the Space m

Using the Green–Samoilenko function, we construct a bounded Frechét-differentiable semiinvariant manifold for a nonlinear system of difference equations in a Banach space of bounded number sequences.     

Weak Solutions of a Generalized Boussinesq System

We study the convergence of homoclinic orbits and heteroclinic orbits in the dynamical system governing traveling wave solutions of a perturbed Boussinesq systems modeling two-directional propagation of water waves. Nonanalytic weak solutions are found to be limits of these orbits, including compactons, peakons, and rampons, as well as infinitely many mesaons occurring at the same fixed point in the dynamical system. Singularities of solitary wave solutions in the system are also studied to understand the important impact of both linear and nonlinear dispersion terms on the regularity of these solutions.     

On the Construction of a Solution of a Degenerate Linear System of Differential Equations in the Neighborhood of an Irregular Singular Point

We investigate the asymptotics of the general solution of the linear system of differential equations with irregular singular point