Global well-posedness for the viscous Camassa-Holm equation

We consider the initial value problem (IVP) of the Camassa-Holm equation with viscosity. We established global solution for the IVP with u₀∈L²(R). This result improves the previous results.     

Global well-posedness for the Benjamin equation in low regularity

In this paper we consider the initial value problem of the Benjamin equation

     

Global well-posedness for the Cauchy problem of the viscous Degasperis-Procesi equation

We establish the local well-posedness for the viscous Degasperis-Procesi equation. We show that the blow-up phenomena occurs in finite time. Moreover, applying the energy identity, we obtain a global existence result in the energy space.     

Low regularity solutions for the wave map equation into the 2-D sphere

A class of weak wave map solutions with initial data in Sobolev space of order s<1 is studied. A non uniqueness result is proved for the case, when the target manifold is a two dimensional sphere. Using an equivariant wave map ansatz a family of self - similar solutions is constructed. This construction enables one to show ill - posedness of the inhomogeneous Cauchy problem for wave maps.Mathematics Subject Classification (2000): 35L70, 58J45.in final form: 17 January 2003     

Global well-posedness for strongly damped viscoelastic wave equation

We study the initial boundary value problem for the nonlinear viscoelastic wave equation with strong damping term and dispersive term. By introducing a family of potential wells we not only obtain the invariant sets, but also prove the existence and nonexistence of global weak solution under some conditions with low initial energy. Furthermore, we establish a blow-up result for certain solutions with arbitrary positive initial energy (high energy case)     

Low regularity solutions for the Kadomtsev-Petviashvili I equation

In this paper we obtain local in time existence and (suitable) uniqueness and continuous dependence for the KP-I equation for small data in the intersection of the energy space and a natural weighted L ² space. An erratum to this article is available at .     

Sharp local well-posedness for a fifth-order shallow water wave equation

In this paper we prove that the already-established local well-posedness in the range s>−5/4 of the Cauchy problem with an initial Hs(R) data for a fifth-order shallow water wave equation is extendable to s=−5/4 by using the space. This is sharp in the sense that the ill-posedness in the range s<−5/4 of this initial value problem is already known.     

Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems

We prove low regularity global well-posedness for the 1d Zakharov system and the 3d Klein-Gordon-Schrödinger system, which are systems in two variables and . The Zakharov system is known to be locally well-posed in and the Klein-Gordon-Schrödinger system is known to be locally well-posed in . Here, we show that the Zakharov and Klein-Gordon-Schrödinger systems are globally well-posed in these spaces, respectively, by using an available conservation law for the norm of and controlling the growth of via the estimates in the local theory.

     

On the well-posedness of derichlet problems for the many-dimensional wave equation and lavrent’ev-bitsadze equation

We prove the unique solvability of the Dirichlet problems for the many-dimensional wave equation and Lavrent’ev-Bitsadze equation.     

Global well-posedness for the KP-I equation

We prove the global well-posedness of the KP-I equation in for data of arbitrary size in a suitable Sobolev class. We use a compactness method to get local solutions. We extend the solutions thanks to some conservation laws, an almost conserved quantity and Strichartz estimates. Received: 5 April 2001 / Revised version: 22 October 2001 / Published online: 17 June 2002     

Sharp well-posedness for the Benjamin equation

Having the ill-posedness in the range s<−3/4 of the Cauchy problem for the Benjamin equation with an initial H^s(R) data, we prove that the already-established local well-posedness in the range s>−3/4 of this initial value problem is extendable to s=−3/4 and also that such a well-posed property is globally valid for s∈[−3/4,∞).     

Global well-posedness,scattering and blow-up for the energy-critical focusing non-linear wave equation

We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space H ¹ than the one of W, then we have global well-posedness and scattering. If the norm is larger than the one of W, then we have break-down in finite time.     

Low regularity solutions to a gently stochastic nonlinear wave equation in nonequilibrium statistical mechanics

We consider a system of stochastic partial differential equations modeling heat conduction in a non-linear medium. We show global existence of solutions for the system in Sobolev spaces of low regularity, including spaces with norm beneath the energy norm. For the special case of thermal equilibrium, we also show the existence of an invariant measure (Gibbs state).     

Modified low regularity well-posedness for the one-dimensional Dirac-Klein-Gordon system

The 1D Cauchy problem for the Dirac-Klein-Gordon system is shown to be locally well-posed for low regularity Dirac data in and wave data in for under certain assumptions on the parameters r and s, where , generalizing the results for p = 2 by Selberg and Tesfahun. Especially we are able to improve the results from the scaling point of view with respect to the Dirac part.      

Global attractors for the viscous hyperelastic‐rod wave equation

We consider the viscous hyperelastic‐rod wave equation subject to an external force, where the viscous term is given by second order differential operator in divergence form. Under some mild assumptions on the viscous term, first, we establish the global well‐posedness in both the periodic case and the case of the whole line, afterwards, we show the existence of global attractors for the two cases, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

Global well-posedness for the fifth-order mKdV equation

We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg–de Vries equation in Sobolev spaces H~s(R) for s-(3/(22)).The main approach is the"I-method"together with the multilinear multiplier analysis.