Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

We establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C^1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C^1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C^1,α, provided that the hyperbolic phase is close in C^1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C^2,α, the free boundary is C^2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.     

Existence of Multiple Positive Solutions of Inhomogeneous Semi–Linear Elliptic Problems Involving Critical Exponents ∗

We consider the obstacle problem for the degenerate Monge-Ampére equation. We prove the existence of the greatest viscosity sub-solution u(x) below a given obstacle φ(x), and its C ^{1, 1}-regularity which is optimal. Then the solution satisfies the concave uniformly elliptic equation if it doesn't touch the obstacle. We use the author's previous work to show the C ^{1, α}-regularity of the free boundary, ?{u(x) = φ(x)}. Finally, we discuss the stability of this free boundary.     

On an integral equation approach for the exterior Robin problem for the Helmholtz equation

A boundary integral equation for the exterior Robin problem for Helmholtz's equation is analyzed in this paper. This integral operator is not compact. A proof based on a suitable regularization of this integral operator and the Fredholm alternative for the regularized compact operator was given by other authors. In this paper, we will give a direct existence and uniqueness proof for the boundary non-compact integral equation in the space settings C^1,λ(S) and C^0,λ(S), where S is a closed bounded smooth surface.     

On the steady viscous flow of a nonhomogeneous asymmetric fluid

We consider a boundary value problem for the system of equations describing the stationary motion of a viscous nonhomogeneous asymmetric fluid in a bounded planar domain having a C ² boundary. We use a stream-function formulation after the manner of Frolov (Math Notes 53(5–6):650–656, 1993) in which the fluid density depends on the stream-function by means of another function determined by the boundary conditions. This allows for dropping some of the equations, most notably the continuity equation. We show existence to the resulting system using Galerkin method.     

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C^2,α$C^{2,\alpha }$-estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.     

An existence and uniqueness result of a nonlinear two-dimensional elliptic boundary value problem

We consider a boundary value problem for the generalized two-dimensional flow equation Δφ = Δφ · h for h a C^α vector field, where the speed is prescribed on a part of the boundary. By using Bers theory combined with elliptic operator theory in nonsmooth domains, we show existence and uniqueness of a C^2,α solution with nonvanishing gradient, and we find positive lower and upper bounds for |Δφ| along with C^2,α estimates of φ, in terms of the C^α and L^∞ norms of h. ©1995 John Wiley & Sons, Inc.     

Weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities and nonlocal free energies

We consider a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions and prove existence of weak solutions for it. In contrast to earlier contributions, we study a model with a singular nonlocal free energy, which controls the H^α/2-norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization.     

A proof for the Burton and Miller integral equation approach for the Helmholtz equation

The A. J. Burton and G. F. Miller integral equation formulation for the exterior Neumann problem for the Helmholtz equation [Proc. Roy. Soc. London Ser. A323 (1971), 201–210] is one of the most important integral equation approaches in that area. However, the kind of space settings they are working with is not clear. Evidently, the Fredholm integral equation of the second kind which they deduced is not well defined on the usual C(S) or L²(S), where S is a closed bounded smooth surface. In this paper, appropriate space settings are found and a rigorous existence and uniqueness proof for their integral equation formulation is given.     

On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains

The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian (−Δ)^α/2, 0<α<2, in a bounded C^1,1 domain D in Rⁿ. In particular, we introduce a new Kato class K_α(D) and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian.     

Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Let E be a finite-dimensional Banach space, let C⁰(R; E) be a Banach space of functions continuous and bounded on R and taking values in E; let K:C ⁰(R ,E) → C ⁰(R, E) be a c-continuous bounded mapping, let A: E → E be a linear continuous mapping, and let h ∈ C ⁰(R, E). We establish conditions for the existence of bounded solutions of the nonlinear equation

Attractor for the viscous two-component Camassa-Holm equation

This paper is concerned with the attractor for a viscous two-component generalization of the Camassa-Holm equation subject to an external force, where the viscosity term is given by a second order differential operator. The global existence of solution to the viscous two-component Camassa-Holm equation with the periodic boundary condition is studied. We obtain the compact and bounded absorbing set and the existence of the global attractor in H²×H² for the viscous two-component Camassa-Holm equation by uniform prior estimate and many inequalities.     

Nodal solutions for singularly perturbed equations with critical exponential growth

The existence and concentration behavior of nodal solutions are established for the equation −?²Δu+V(z)u=f(u) in Ω, where Ω is a domain in R², not necessarily bounded, V is a positive Hölder continuous function and f∈C¹ is an odd function having critical exponential growth.     

A General Approximation Scheme for Attractors of Abstract Parabolic Problems

We consider semilinear problems of the form u′ = Au + f(u), where A generates an exponentially decaying compact analytic C ₀-semigroup in a Banach space E, f:E ^α → E is differentiable globally Lipschitz and bounded (E ^α = D((?A)^α) with the graph norm). Under a very general approximation scheme, we prove that attractors for such problems behave upper semicontinuously. If all equilibrium points are hyperbolic, then there is an odd number of them. If, in addition, all global solutions converge as t → ±∞, then the attractors behave lower semicontinuously. This general approximation scheme includes finite element method, projection and finite difference methods. The main assumption on the approximation is the compact convergence of resolvents, which may be applied to many other problems not related to discretization.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

A C0 interior penalty discontinuous Galerkin Method for fourth‐order total variation flow. II: Existence and uniqueness

We prove the existence and uniqueness of a solution of a C⁰ Interior Penalty Discontinuous Galerkin (C⁰ IPDG) method for the numerical solution of a fourth‐order total variation flow problem that has been developed in part I of the paper. The proof relies on a nonlinear version of the Lax‐Milgram Lemma. It requires to establish that the nonlinear operator associated with the C⁰ IPDG approximation is Lipschitz continuous and strongly monotone on bounded sets of the underlying finite element space.     

Pullback attractors for a semilinear heat equation on time-varying domains

The existence of a pullback attractor is established for the nonautonomous dynamical system generated by the weak solutions of a semilinear heat equation on time-varying domains with homogeneous Dirichlet boundary conditions. It is assumed that the spatial domains Ot in RN are obtained from a bounded base domain O by a C²-diffeomorphism, which is continuously differentiable in the time variable, and are contained, in the past, in a common bounded domain.     

Global weak solutions to one-dimensional compressible viscous hydrodynamic equations

In this article, we are concerned with the global weak solutions to the 1D compressible viscous hydrodynamic equations with dispersion correction δ~2ρ((φ(ρ))xxφ′(ρ))x withφ(ρ) = ρα. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial conditions. The diffusion term εu xx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in[1](α =1/2) to 0 α≤1. In addition, we perform the limit ε→ 0 with respect to 0 α≤1/2.     

Halbgruppen und semilineare Anfangs-Randwertprobleme

A semilinear parabolic initial-boundary-value problem of order 2m in a possibly unbounded domain Ωx(O,T), Ω?Rⁿ, is considered within the framework of the L_p-and C^α-theory. In the first case a proof is given of the existence of a “strict” solution of the corresponding evolution equation. In the second case one can guarantee a classical solution, provided the homogeneous linear parabolic equation has a unique classical solution. Only local solvability is considered. The nonlinearity is a Hölder-continuous function of the derivatives up to the order 2m-1 of the unknown solution. The principal tool is the semigroup-theory in L_p(Ω) as well as in C^α( \(\bar \Omega \) ). In the latter case the semigroup is not strongly continuous, but it has sufficiently good properties to use it for existence proofs of classical solutions.     

The global attractor of the viscous Fornberg–Whitham equation

This paper aims to present a proof of the existence of the attractor for the one-dimensional viscous Fornberg–Whitham equation. In this paper, the global existence of solution to the viscous Fornberg–Whitham equation in L² under the periodic boundary conditions is studied. By using the time estimate of the Fornberg–Whitham equation, we get the compact and bounded absorbing set and the existence of the global attractor for the viscous Fornberg–Whitham equation.