Energy critical fourth-order Schrödinger equations with subcritical perturbations

In this paper, we study the global well-posedness and scattering theory of an 8-D cubic nonlinear fourth-order Schrödinger equation, which is perturbed by a subcritical nonlinearity. We utilize the strategies in Tao et al. (2007) [16] and Zhang (2006) [17] to obtain when the cubic term is defocusing, the solution is always global no matter what the sign of the subcritical perturbation term is. Moreover, scattering will occur either when the pertubation is defocusing and 1<p<2 or when the mass of the solution is small enough and 1≤p<2.     

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.     

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.     

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.     

Multilinear estimates for periodic KdV equations, and applications

We prove an endpoint multilinear estimate for the X^s,b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm.     

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.     

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 critical Burgers equation in critical Besov spaces

We make use of the method of modulus of continuity [A. Kiselev, F. Nazarov, R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 (2008) 211-240] and Fourier localization technique [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167-185] to prove the global well-posedness of the critical Burgers equation t_∂u+ux_∂u+Λu=0 in critical Besov spaces with p∈[1,∞), where .     

Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: Exponential and polynomial stabilization

Results of exponential/polynomial decay rates of the energy in L²-level, related to the cubic nonlinear Schrödinger equation with localized damping posed on the whole real line, will be established in this paper. We use Kato's theory and a priori estimates to obtain the result of global well-posedness and we determine exponential/polynomial stabilization combining the ideas of unique continuation due to Zhang, semigroup property and Komornik's approach.     

On the low regularity of the fifth order Kadomtsev-Petviashvili I equation

We consider the fifth order Kadomtsev-Petviashvili I (KP-I) equation as , while α∈R. We introduce an interpolated energy space Es to consider the well-posedness of the initial value problem (IVP) of the fifth order KP-I equation. We obtain the local well-posedness of IVP of the fifth order KP-I equation in Es for 0<s?1. To obtain the local well-posedness, we present a bilinear estimate in the Bourgain space in the framework of the interpolated energy space. It crucially depends on the dyadic decomposed Strichartz estimate, the fifth order dispersive smoothing effect and maximal estimate.     

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,∞).     

On the classification of minimal mass blowup solutions of the focusing mass-critical Hartree equation

Consider the focusing mass-critical nonlinear Hartree equation iut+Δu=−(⁻²|⋅|∗²|u|)u for spherically symmetric initial data with ground state mass M(Q) in dimension d?5. We show that any global solution u which does not scatter must be the solitary wave eitQ up to phase rotation and scaling.     

Long range scattering for the modified Schrödinger map in two space dimensions

We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schrödinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we determine the asymptotic behaviour in time of the solutions thereby obtained. As a by product, we obtain global existence for small data in Hk∩FHk with k>1. We also solve the Cauchy problem with infinite initial time, namely we construct solutions defined in a neighborhood of infinity in time, with prescribed asymptotic behaviour of the previous type.     

Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework

We investigate global strong solution to a 3-dimensional parabolic–hyperbolic system arising from the Keller–Segel model. We establish the global well-posedness and asymptotic behavior in the energy functional setting. Precisely speaking, if the initial difference between cell density and its mean is small in L²$L^2$, and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in H¹$H^1$, then they remain to be small in L²×H¹$L^2\times H^1$ for all time. Moreover, if the mean value of the initial cell density is smaller than some constant, then the cell density approaches its initial mean and the chemical concentration decays exponentially to zero as t goes to infinity. The proof relies on an application of Fourier analysis to a linearized parabolic–hyperbolic system and the smoothing effect of the cell density and the damping effect of the chemical concentration.     

The focusing energy-critical Hartree equation

We consider the focusing energy-critical nonlinear Hartree equation iut+Δu=−(⁻⁴|x|∗²|u|)u. We proved that if a maximal-lifespan solution u:I×Rd→C satisfies supt∈I‖∇u(t)_‖2<‖∇W_‖2, where W is the static solution of the equation, then the maximal-lifespan I=R, moreover, the solution scatters in both time directions. For spherically symmetric initial data, similar result has been obtained in [C. Miao, G. Xu, L. Zhao, Global wellposedness, scattering and blowup for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math., in press]. The argument is an adaptation of the recent work of R. Killip and M. Visan [R. Killip, M. Visan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, preprint] on energy-critical nonlinear Schrödinger equations.     

On the Cauchy problem of a weakly dissipative μ-Hunter–Saxton equation

In this paper, we study the Cauchy problem of a weakly dissipative μ-Hunter–Saxton equation. We first establish the local well-posedness for the weakly dissipative μ-Hunter–Saxton equation by Kato's semigroup theory. Then, we derive the precise blow-up scenario for strong solutions to the equation. Moreover, we present some blow-up results for strong solutions to the equation. Finally, we give two global existence results to the equation.     

Rough solutions for the periodic Schrödinger-Korteweg-de Vries system

We prove two new mixed sharp bilinear estimates of Schrödinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schrödinger-Kortweg-de Vries (NLS-KdV) system in the periodic setting. Our lowest regularity is H1/4×L², which is somewhat far from the naturally expected endpoint L²×H−1/2. This is a novel phenomena related to the periodicity condition. Indeed, in the continuous case, Corcho and Linares proved local well-posedness for the natural endpoint .Nevertheless, we conclude the global well-posedness of the NLS-KdV system in the energy space H¹×H¹ using our local well-posedness result and three conservation laws discovered by M. Tsutsumi.     

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 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].