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 (∗).     

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 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.     

Blow-up behavior outside the origin for a semilinear wave equation in the radial case

We consider the semilinear wave equation in the radial case with conformal subcritical power nonlinearity. If we consider a blow-up point different from the origin, then we exhibit a new Lyapunov functional which is a perturbation of the one-dimensional case and extend all our previous results known in the one-dimensional case. In particular, we show that the blow-up set near non-zero non-characteristic points is of class C¹, and that the set of characteristic points is made of concentric spheres in finite number in for any R>1.     

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.     

Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

This paper is devoted to studying a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under the suitable conditions, we prove that any weak solutions with negative initial energy will blow up in finite time. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

The hyperbolic mean curvature flow

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.     

Global uniqueness and reconstruction for the multi-channel Gel?fand–Calderón inverse problem in two dimensions

We study the multi-channel Gel?fand–Calderón inverse problem in two dimensions, i.e. the inverse boundary value problem for the equation −Δψ+v(x)ψ=0, x∈D, where v is a smooth matrix-valued potential defined on a bounded planar domain D. We give an exact global reconstruction method for finding v from the associated Dirichlet-to-Neumann operator. This also yields a global uniqueness results: if two smooth matrix-valued potentials defined on a bounded planar domain have the same Dirichlet-to-Neumann operator then they coincide.     

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.     

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.     

A Reilly inequality for the first Steklov eigenvalue

Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p₁ of Steklov problem on M in terms of the r-th mean curvatures of its boundary ∂M. The upper bound obtained is sharp.     

NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS

In this article, we get non-selfsimilar elementary waves of the conservation laws in another kind of view, which is different from the usual self-similar transformation. The solution has different global structure. This article is divided into three parts. The first part is introduction. In the second part, we discuss non-selfsimilar elementary waves and their interactions of a class of twodimensional conservation laws. In this case, we consider the case that the initial discontinuity is parabola with u＋ 〉 0, while explicit non-selfsirnilar rarefaction wave can be obtained. In the second part, we consider the solution structure of case u＋ 〈 0. The new solution structures are obtained by the interactions between different elementary waves, and will continue to interact with other states. Global solutions would be very different from the situation of one dimension.     

Remark on the global existence and large time asymptotics of solutions for the quadratic NLS

We study the global in time existence of small solutions to the nonlinear Schrödinger equation with quadratic interactions (0.1) We prove that if the initial data u₀ satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore, we prove the existence of the usual scattering states and find the large time asymptotics of the solutions.     

Blow up and global solutions for a quasilinear riser problem

In this paper, we consider an initial–boundary value problem to a riser vibrating with dissipative term in the equation. It is proved that under suitable conditions that the solution with a negative initial energy blows up in finite time. And we show that the solution with a nonnegative initial energy is global.     

Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps

We consider non-local linear Schrödinger-type critical systems of the type(1) where Ω is antisymmetric potential in L²(R,so(m)), v is an Rm valued map and Ωv denotes the matrix multiplication. We show that every solution v∈L²(R,Rm) of (1) is in fact in , for every 2?p<+∞, in other words, we prove that the system (1) which is a-priori only critical in L² happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the regularity of weak 1/2-harmonic maps into C² compact sub-manifolds without boundary.     

The Third Problem for the Laplace Equation with a Boundary Condition from L p

The third problem for the Laplace equation is studied on an open set with Lipschitz boundary. The boundary condition is in L_p and it is fulfilled in the sense of the nontangential limit. The existence and the uniqueness of a solution is proved and the solution is expressed in the form of a single layer potential. For domains with C¹ boundary the explicit solution of the problem is calculated.     

The Baum–Connes conjecture for free orthogonal quantum groups

We prove an analogue of the Baum–Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a γ-element and that γ=1. It follows that free orthogonal quantum groups are K-amenable. We compute explicitly their K-theory and deduce in the unimodular case that the corresponding reduced C^?-algebras do not contain nontrivial idempotents.Our approach is based on the reformulation of the Baum–Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equivalence of compact quantum groups developed by Bichon, De Rijdt and Vaes. This allows us to study the problem in terms of the quantum group SUq(2). The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podle? sphere.     

Convergence of the Dirichlet solutions of the very fast diffusion equation

For any −1<m<0, positive functions f, g and u₀≥0, we prove that under some mild conditions on f, g and u₀ as R→∞ the solution u^R of the Dirichlet problem u_t=(u^m/m)_xx in (−R,R)×(0,∞), u(R,t)=(f(t)|m|R)^1/m, u(−R,t)=(g(t)|m|R)^1/m for all t>0, u(x,0)=u₀(x) in (−R,R), converges uniformly on every compact subset of R×(0,T) to the solution of the equation u_t=(u^m/m)_xx in R×(0,T), u(x,0)=u₀(x) in R, which satisfies some mass loss formula on (0,T) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) [15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.     

Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization

The following reaction-diffusion system in spatially non-homogeneous almost-periodic media is considered in a bounded domain : (1)∂_tu=Au−f(u)+g, u|_∂Ω=0. Here u=(u¹,…,u^k) is an unknown vector-valued function, f is a given nonlinear interaction function and the second order elliptic operator A has the following structure: where a_ij^l(y) are given almost-periodic functions. We prove that, under natural assumptions on the nonlinear term f(u), the longtime behavior of solutions of (1) can be described in terms of the global attractor of the associated dynamical system and that the attractors  , 0<<₀1, converge to the attractor of the homogenized problem (1) as →0. Moreover, in the particular case of periodic media, we give explicit estimates for the distance between the non-homogenized and the homogenized attractors in terms of the parameter .     

Existence, uniqueness and asymptotic stability of time periodic traveling waves for a periodic Lotka–Volterra competition system with diffusion

We study the existence, uniqueness, and asymptotic stability of time periodic traveling wave solutions to a periodic diffusive Lotka–Volterra competition system. Under certain conditions, we prove that there exists a maximal wave speed c^? such that for each wave speed c?c^?, there is a time periodic traveling wave connecting two semi-trivial periodic solutions of the corresponding kinetic system. It is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co-moving frame coordinate. We also show that the traveling wave solutions with wave speed c<c^? are asymptotically stable in certain sense. In addition, we establish the nonexistence of time periodic traveling waves for nonzero speed c>c^?.