This current paper is devoted to the Cauchy problem for higher order dispersive equation ut＋δx^2n＋1u=δx（uδx^nu）＋δx^n-1（ux^2）, n ≥ 2, n ∈ N^＋. Ut By using Besov-type spaces, we prove that the associated ...This current paper is devoted to the Cauchy problem for higher order dispersive equation ut＋δx^2n＋1u=δx（uδx^nu）＋δx^n-1（ux^2）, n ≥ 2, n ∈ N^＋. Ut By using Besov-type spaces, we prove that the associated problem is locally well-posed in H（-n/2＋3/4,-1/2n）. The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H^（s,a）（R） with s〈-n/2＋3/4 and all a∈R.展开更多
This paper offers a variant of a proof of a borderline Bourgain-Brezis Sobolev embedding theorem on R^n. The authors use this idea to extend the result to real hyperbolic spaces H^n.
We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R^（1＋1）. ...We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R^（1＋1）. We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schr¨odinger data in H^（s_1） and wave data in H^（s_2） × H^（s_2-1）for 3/4- α ＆lt; s_1≤0 and-1/2 ＆lt; s_2 ＆lt; 3/2, where α is the fractional power of Laplacian which satisfies 3/4 ＆lt; α≤1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s_1 = 0 and-1/2 ＆lt; s_2 ＆lt; 1/2 by using the conservation law for the L^2 norm of u.展开更多
We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg-de Vries equation in Sobolev spaces H^3（R） for s〉-3/22.The main approach is the ＂I-method＂ together with the multilinear ...We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg-de Vries equation in Sobolev spaces H^3（R） for s〉-3/22.The main approach is the ＂I-method＂ together with the multilinear multiplier analysis.展开更多
We study nonlinear Schr¨odinger equations on Zoll manifolds with nonlinear growth of the odd order.It is proved that local uniform well-posedness are valid in the Hs-subcritical setting according to the scaling i...We study nonlinear Schr¨odinger equations on Zoll manifolds with nonlinear growth of the odd order.It is proved that local uniform well-posedness are valid in the Hs-subcritical setting according to the scaling invariance, apart from the cubic growth in dimension two. This extends the results by Burq et al.(2005) to higher dimensions with general nonlinearities.展开更多
The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut=iαu_(xx)+βu^(2)u_(x)+γ|u|^(2)u_(x)+i|u|^(2)u on the half-line with inhomogeneous b...The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut=iαu_(xx)+βu^(2)u_(x)+γ|u|^(2)u_(x)+i|u|^(2)u on the half-line with inhomogeneous boundary condition.We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces.Moreover,we show that the nonlinear part of the solution on the half-line is smoother than the initial data.展开更多
基金supported by Natural Science Foundation of China NSFC(11401180 and 11471330)supported by the Young Core Teachers Program of Henan Normal University(15A110033)supported by the Fundamental Research Funds for the Central Universities(WUT:2017 IVA 075)
文摘This current paper is devoted to the Cauchy problem for higher order dispersive equation ut＋δx^2n＋1u=δx（uδx^nu）＋δx^n-1（ux^2）, n ≥ 2, n ∈ N^＋. Ut By using Besov-type spaces, we prove that the associated problem is locally well-posed in H（-n/2＋3/4,-1/2n）. The new ingredient is that we establish some new dyadic bilinear estimates. When n is even, we also prove that the associated equation is ill-posed in H^（s,a）（R） with s〈-n/2＋3/4 and all a∈R.
基金S.Chanillo was partially supported by NSF grant DMS 1201474J.Van Schaftingen was partially supported by the Fonds de la Recherche Scientifique-FNRS+3 种基金P.-L.Yung was partially supported by a Titchmarsh Fellowship at the University of Oxforda junior research fellowship at St.Hilda’s Collegea direct grant for research from the Chinese University of Hong Kong(3132713)an Early Career Grant CUHK24300915 from the Hong Kong Research Grant Council
文摘This paper offers a variant of a proof of a borderline Bourgain-Brezis Sobolev embedding theorem on R^n. The authors use this idea to extend the result to real hyperbolic spaces H^n.
基金supported by National Natural Science Foundation of China (Grant No. 11201498)
文摘We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R^（1＋1）. We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schr¨odinger data in H^（s_1） and wave data in H^（s_2） × H^（s_2-1）for 3/4- α ＆lt; s_1≤0 and-1/2 ＆lt; s_2 ＆lt; 3/2, where α is the fractional power of Laplacian which satisfies 3/4 ＆lt; α≤1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s_1 = 0 and-1/2 ＆lt; s_2 ＆lt; 1/2 by using the conservation law for the L^2 norm of u.
文摘We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg-de Vries equation in Sobolev spaces H^3（R） for s〉-3/22.The main approach is the ＂I-method＂ together with the multilinear multiplier analysis.
基金supported by National Natural Science Foundation of China(Grant Nos.11171033 and 11231006)
文摘We study nonlinear Schr¨odinger equations on Zoll manifolds with nonlinear growth of the odd order.It is proved that local uniform well-posedness are valid in the Hs-subcritical setting according to the scaling invariance, apart from the cubic growth in dimension two. This extends the results by Burq et al.(2005) to higher dimensions with general nonlinearities.
文摘The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut=iαu_(xx)+βu^(2)u_(x)+γ|u|^(2)u_(x)+i|u|^(2)u on the half-line with inhomogeneous boundary condition.We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces.Moreover,we show that the nonlinear part of the solution on the half-line is smoother than the initial data.
