Well posedness for a higher order modified Camassa-Holm equation

We show that the Cauchy problem for a higher order modification of the Camassa-Holm equation is locally well posed for initial data in the Sobolev space Hs(R) for s>s^′, where 1/4?s^′<1/2 and the value of s^′ depends on the order of equation. Employing harmonic analysis methods we derive the corresponding bilinear estimate and then use a contraction mapping argument to prove existence and uniqueness of solutions.     

On the low regularity of the modified Korteweg-de Vries equation with a dissipative term

We consider a dissipative version of the modified Korteweg-de Vries equation ut+uxxx−uxx+x(u³)=0. We prove global well-posedness results on the associated Cauchy problem in the Sobolev spaces Hs(R) for s>−1/4 while for s<−1/2 we prove some ill-posedness issues.     

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,∞).     

Biharmonic nonlinear Schrödinger equation and the profile decomposition

This paper is concerned with the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L²-super-critical nonlinearity. By establishing the profile decomposition of bounded sequences in H²(R^N), the best constant of a Gagliardo-Nirenberg inequality is obtained. Moreover, a sufficient condition for the global existence of the solution to the biharmonic nonlinear Schrödinger equation is given.     

Global well-posedness for the generalized 2D Ginzburg-Landau equation

The local well-posedness for the generalized two-dimensional (2D) Ginzburg-Landau equation is obtained for initial data in Hs(R²)(s>1/2). The global result is also obtained in Hs(R²)(s>1/2) under some conditions. The results on local and global well-posedness are sharp except the endpoint s=1/2. We mainly use the Tao's [k;Z]-multiplier method to obtain the trilinear and multilinear estimates.     

Well-posedness and blow-up phenomena for a higher order shallow water equation

This paper deals with the Cauchy problem for a higher order shallow water equation yt+auxy+buyx=0, where and k=2. The local well-posedness of solutions for the Cauchy problem in Sobolev space Hs(R) with s?7/2 is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are also acquired. Finally, the weak solution for the equation is considered.     

On the well-posedness of the Schrödinger-Korteweg-de Vries system

We prove that the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sobolev spaces L²(R)×H−3/4(R), and Hs(R)×H−3/4(R) (s>−1/16) for the resonant case. The new ingredient is that we use the -type space, introduced by the first author in Guo (2009) [10], to deal with the KdV part of the system and the coupling terms. In order to overcome the difficulty caused by the lack of scaling invariance, we prove uniform estimates for the multiplier. This result improves the previous one by Corcho and Linares (2007) [6].     

On global solution to the Klein-Gordon-Hartree equation below energy space

In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in R³. By making use of Bourgain's method in conjunction with a precise Strichartz estimate of S. Klainerman and D. Tataru, we establish the Hs (s<1) global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the previously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in Hs₀×Hs₀−1 where but not , for this equation that we consider is a subconformal equation in some sense. In doing so a number of nonlinear prior estimates are already established by using Bony's decomposition, flexibility of Klein-Gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate. We establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems that this is the first result on low regularity for this Klein-Gordon-Hartree equation.     

Sharp ill-posedness result for the periodic Benjamin-Ono equation

We prove the discontinuity for the weak L²(T)-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in Hs(T) as soon as s<0 and thus completes exactly the well-posedness result obtained in Molinet (2008) [12].     

Global rough solutions to the critical generalized KdV equation

We prove that the initial value problem (IVP) for the critical generalized KdV equation ut+uxxx+x(u⁵)=0 on the real line is globally well-posed in Hs(R) if s>3/5 with the appropriate smallness assumption on the initial data.     

Small solutions to nonlinear Schrödinger equations in the Sobolev spaces

The local and global well-posedness for the Cauchy problem for a class of nonlinear Schrödinger equations is studied. The global well-posedness of the problem is proved in the Sobolev spaceH ^s=H^s(R ⁿ) of fractional orders>n/2 under the following assumptions. (1) Concerning the Cauchy data ?∈H ^s: ‖?;L ²‖ is relatively small with respect to ‖?;H ^σ‖ for any fixed σ withn/2<σ≤s. (2) Concerning the nonlinearityf: f(u) behaves as a conformal poweru ¹⁺⁴/n near zero and has an arbitrary growth rate at infinity.     

Further regularity results for almost cubic nonlinear Schrödinger equation

We present three results related with the regularity of solutions of the almost cubic NLS. In the first one, following Ozawa’s idea, we establish mass and energy conservation for the solutions without regularizing the initial datum. Our second result is the H^s well-posedness for the Cauchy problem for 0<s<1. Finally, we show that the same solutions are also in some Bourgain spaces for possibly a smaller time interval. In all of our results, the non-local nonlinear term in the equation is shown to act like a cubic nonlinearity on the appropriate Sobolev and Besov spaces.     

Well-posedness and weak rotation limit for the Ostrovsky equation

We consider the Cauchy problem of the Ostrovsky equation. We first prove the time local well-posedness in the anisotropic Sobolev space Hs,a with s>−a/2−3/4 and 0?a?−1 by the Fourier restriction norm method. This result include the time local well-posedness in Hs with s>−3/4 for both positive and negative dissipation, namely for both βγ>0 and βγ<0. We next consider the weak rotation limit. We prove that the solution of the Ostrovsky equation converges to the solution of the KdV equation when the rotation parameter γ goes to 0 and the initial data of the KdV equation is in L². To show this result, we prove a bilinear estimate which is uniform with respect to γ.     

Global well-posedness of the Cauchy problem of the fifth-order shallow water equation

By using the I-method, we prove that the Cauchy problem of the fifth-order shallow water equation is globally well-posed in the Sobolev space Hs(R) provided .     

Global solutions and blow-up phenomena to a shallow water equation

A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. The local well-posedness of solutions for the nonlinear equation in the Sobolev space Hs(R) with is developed. Provided that does not change sign, u₀∈Hs () and u₀∈L¹(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x)∈C([0,∞);Hs(R))∩C¹([0,∞);Hs−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired.     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant

In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v₀(x),u₀(x)), the corresponding initial-boundary value problem admits a unique global smooth solution (v(x,t),u(x,t)) and such a solution tends time-asymptotically, at the Lp (2?p?∞) optimal decay rates, to the corresponding nonlinear diffusion wave which satisfies (1.9) provided the corresponding prescribed initial error function (V₀(x),U₀(x)) lies in (H³(R⁺)∩L¹(R⁺))×(H²(R⁺)∩L¹(R⁺)).     

Quasilinear asymptotically periodic Schrödinger equations with subcritical growth

In this article the existence of one solution for a class of asymptotically periodic equations in the euclidean space is established. The basic tools employed here are the Mountain Pass Theorem and the Concentration-Compactness Principle. By using a change of variable, the quasilinear equation is reduced to a semilinear equation, whose respective associated functional is well defined in H¹(R^N) and satisfies the geometric hypotheses of the Mountain Pass Theorem.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

Solution of Quasilinear Stochastic Problems in Abstract Colombeau Algebras

The main object of study is the stochastic Cauchy problem for a quasilinear equation with random disturbances in the form of a Hilbert-valued white noise process and with an operator generating an integrated semigroup in the space L²(R). We use the Colombeau theory of multiplication of distributions to introduce an abstract stochastic factor algebra and construct an approximate solution of the problem in this algebra.     

Global existence for 2D nonlinear Schrödinger equations via high-low frequency decomposition method

We study global existence of solutions for the Cauchy problem of the nonlinear Schrödinger equation iut+Δu=|u|^2mu in the 2 dimension case, where m is a positive integer, m?2. Using the high-low frequency decomposition method, we prove that if then for any initial value φ∈Hs(R²), the Cauchy problem has a global solution in C(R,Hs(R²)), and it can be split into u(t)=eitΔφ+y(t), with y∈C(R,H¹(R²)) satisfying , where ? is an arbitrary sufficiently small positive number.