The relaxation-time limit in the compressible Euler–Maxwell equations

The aim of this paper is to study multidimensional Euler–Maxwell equations for plasmas with short momentum relaxation time. The convergence for the smooth solutions to the compressible Euler–Maxwell equations toward the solutions to the smooth solutions to the drift–diffusion equations is proved by means of the Maxwell iteration, as the relaxation time tends to zero. Meanwhile, the formal derivation of the latter from the former is justified.     

Ill-posedness of nonlocal Burgers equations

Nonlocal generalizations of Burgers equation were derived in earlier work by Hunter [J.K. Hunter, Nonlinear surface waves, in: Current Progress in Hyberbolic Systems: Riemann Problems and Computations, Brunswick, ME, 1988, in: Contemp. Math., vol. 100, Amer. Math. Soc., 1989, pp. 185–202], and more recently by Benzoni-Gavage and Rosini [S. Benzoni-Gavage, M. Rosini, Weakly nonlinear surface waves and subsonic phase boundaries, Comput. Math. Appl. 57 (3–4) (2009) 1463–1484], as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage [S. Benzoni-Gavage, Local well-posedness of nonlocal Burgers equations, Differential Integral Equations 22 (3–4) (2009) 303–320] under an appropriate stability condition originally pointed out by Hunter. In this article, it is shown that the latter condition is not only sufficient for well-posedness in Sobolev spaces but also necessary. The main point of the analysis is to show that when the stability condition is violated, nonlocal Burgers equations reduce to second order elliptic PDEs. The resulting ill-posedness result encompasses various cases previously studied in the literature.     

Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators

In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDE of various types such as porous medium equations, reaction–diffusion equations, the generalized Burgers equation, the Navier–Stokes equation, the 3D Leray-α model and the p-Laplace equation with non-monotone perturbations.     

Gevrey regularity of subelliptic Monge–Ampère equations in the plane

In this paper, we establish the Gevrey regularity of solutions for a class of degenerate Monge–Ampère equations in the plane. Under the assumptions that one principal entry of the Hessian is strictly positive and the coefficient of the equation is degenerate with appropriately finite type degeneracy, we prove that the solution of the degenerate Monge–Ampère equation will be smooth in Gevrey classes.     

The existence and asymptotics of solutions for the Abelian Chern–Simons system with two Higgs fields and two gauge fields

In this paper, we study a system of elliptic equations in R² which arises from the self-dual equations for the Abelian Chern–Simons system with two Higgs fields and two gauge fields. We provide a new proof for the existence of topological solutions by constructing explicit supersolutions and subsolutions. We also study the asymptotic behavior of condensate solutions on the torus. It is shown that the maximal solutions converge uniformly to zero away from the vortex points, and the convergence rate is computed.     

A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations

This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities.     

Decay estimates for quasi-linear evolution equations

We consider global strong solutions of the quasi-linear evolution equations (1.1) and (1.2) below, corresponding to sufficiently small initial data, and prove some stability estimates, as t→+∞, that generalize the corresponding estimates in the linear case.     

Superharmonic functions are locally renormalized solutions

We show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent. As an application we characterize singular solutions of multidimensional Riccati type partial differential equations.     

Pseudo-almost periodic viscosity solutions of second-order nonlinear parabolic equations

In this paper we generalize the comparison result of Bostan and Namah (2007) [8] to the second-order parabolic case and prove two properties of pseudo-almost periodic functions; then by using Perron’s method we prove the existence and uniqueness of time pseudo-almost periodic viscosity solutions of second-order parabolic equations under usual hypotheses.     

A parallel four step domain decomposition scheme for coupled forward–backward stochastic differential equations

Motivated by the idea of imposing paralleling computing on solving stochastic differential equations (SDEs), we introduce a new domain decomposition scheme to solve forward–backward stochastic differential equations (FBSDEs) parallel. We reconstruct the four step scheme in Ma et al. (1994) [1] and then associate it with the idea of domain decomposition methods. We also introduce a new technique to prove the convergence of domain decomposition methods for systems of quasilinear parabolic equations and use it to prove the convergence of our scheme for the FBSDEs.     

Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms

We prove some symmetry property for equations with Hardy terms in cones, without any assumption at infinity. We also show symmetry property and nonexistence of entire solutions of some elliptic systems with Hardy weights.     

Compressible navier-stokes-poisson equations

This is a survey paper on the study of compressible Navier-Stokes-Poisson equations. The emphasis is on the long time behavior of global solutions to multi-dimensional compressible Navier-Stokes-Poisson equations, and the optimal decay rates for both unipolar and bipolar compressible Navier-Stokes-Poisson equations are discussed.     

Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.     

Local solutions for a coupled system of Kirchhoff type

We investigate the existence of local solutions of the following coupled system of Kirchhoff equations subject to nonlinear dissipation on the boundary: (∗) Here {Γ₀,Γ₁} is an appropriate partition of the boundary Γ of Ω and ν(x), the outer unit normal vector at x∈Γ₁.By applying the Galerkin method with a special basis for the space where lie the approximations of the initial data, we obtain local solutions of the initial-boundary value problem for (∗).     

Liouville type theorems for the Euler and the Navier–Stokes equations

We prove Liouville type theorems for weak solutions of the Navier–Stokes and the Euler equations. In particular, if the pressure satisfies p∈L¹(0,T;L¹(RN)) with , then the corresponding velocity should be trivial, namely v=0 on RN×(0,T). In particular, this is the case when p∈L¹(0,T;Hq(RN)), where Hq(RN), q∈(0,1], the Hardy space. On the other hand, we have equipartition of energy over each component, if p∈L¹(0,T;L¹(RN)) with . Similar results hold also for the magnetohydrodynamic equations.     

Mild solutions of functional semilinear evolution Volterra integrodifferential equations on an unbounded interval

In this paper, we study the existence of mild solutions for initial value problems for semilinear Volterra integrodifferential equations in a Banach space. The arguments are based on the concept of measure of noncompactness in Fréchet space and the Tikhonov fixed point theorem.     

On the continuation of solutions for elliptic equations in two variables

We consider nonlinear elliptic differential equations of second order in two variables

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. Supposing analyticity of F, we prove analyticity of the real solution z=z(x,y) in the open set Ω. Furthermore, we show that z may be continued as a real analytic solution for F=0 across the real analytic boundary arc Γ∂Ω, if z satisfies one of the boundary conditions z= or z_n=ψ(x,y,z,z_t) on Γ with real analytic functions and ψ, respectively (z_n denotes the derivative of z w.r.t. the outer normal n on Γ and z_t its derivative w.r.t. the tangent). The proof is based on ideas of H. Lewy combined with a uniformization method. Studying quasilinear equations, we get somewhat better results concerning the initial regularity of the given solution and a little more insight.     

Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic–parabolic systems

Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic–parabolic systems including the Navier–Stokes equations of compressible gas, and magnetohydrodynamics with inflow or outflow boundary conditions, establishing that linear and nonlinear stability are both equivalent to an Evans function, or generalized spectral stability, condition. The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in particular, it has been shown by Costanzino, Humpherys, Nguyen, and Zumbrun to hold for sufficiently large-amplitude layers for isentropic ideal gas dynamics, with general adiabiatic index γ?1. Together with these previous results, our results thus give nonlinear stability of large-amplitude isentropic boundary layers, the first such result for compressive (“shock-type”) layers in other than the nearly-constant case. The analysis, as in the strictly parabolic case, proceeds by derivation of detailed pointwise Green function bounds, with substantial new technical difficulties associated with the more singular, hyperbolic behavior in the high-frequency/short time regime.     

Strong trajectory attractors for dissipative Euler equations

The 2D Euler equations with periodic boundary conditions and extra linear dissipative term Ru, R>0 are considered and the existence of a strong trajectory attractor in the space is established under the assumption that the external forces have bounded vorticity. This result is obtained by proving that any solution belonging the proper weak trajectory attractor has a bounded vorticity which implies its uniqueness (due to the Yudovich theorem) and allows to verify the validity of the energy equality on the weak attractor. The convergence to the attractor in the strong topology is then proved via the energy method.     

On shock generation for Hamilton-Jacobi equations

The subject of this paper is the generation of singularities of solutions of Hamilton-Jacobi equations set in (0, ∞) × ? fordataofclass C∞. Shockwaves originate from conjugate points. To show sharpness of a known Hausdorff estimate, an example is given in which the set of conjugate, regular points includes uncountably many affine subspaces of dimension n − 1.