The weUposedness problem for an anisotropic incompressible viscous fluid in R3, ro- tating around a vector B（t, x） ：= （b1 （t, x）, b2 （t, x）, b3 （t, x））, is studied. The global wellposedness in the homogeneo...The weUposedness problem for an anisotropic incompressible viscous fluid in R3, ro- tating around a vector B（t, x） ：= （b1 （t, x）, b2 （t, x）, b3 （t, x））, is studied. The global wellposedness in the homogeneous case （B = e3） with sufficiently fast rotation in the space B0,1/2 is proved. In the inhomogeneous case （B = B（t, xh））, the global existence and uniqueness of the solution in B0,1/2 are obtained, provided that the initial data are sufficient small compared to the horizontal viscosity. Furthermore, we obtain uniform local existence and uniqueness of the solution in the x same function space. We also obtain propagation of the regularity in B2,11/2 under the additional assumption that B depends only on one horizontal space variable.展开更多
We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity.The primary goal of this paper is to understand how multi-...We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity.The primary goal of this paper is to understand how multi-component equations,nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons.We establish the local well-posedness of the Cauchy problem in Besov spaces B^s/p,r with 1 p,r+∞,s>max{1+1/p,3/2}and Sobolev spaces H^s(R)with s>3/2,and the method is based on the estimates for transport equations and new invariant properties of the system.Furthermore,the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied.A blow-up criterion on solutions of the Cauchy problem is demonstrated.In addition,we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line,and the single-peaked solitons on the circle,which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.展开更多
In this paper we prove that the Schrodinger-Boussinesq system with solution(u,v,(-∂xx)-^(2/1)vt)is locally wellposed in H^(s)×H^(s)×Hs^(-1),s≥-1/4.The local wellposedness is obtained by the transformation f...In this paper we prove that the Schrodinger-Boussinesq system with solution(u,v,(-∂xx)-^(2/1)vt)is locally wellposed in H^(s)×H^(s)×Hs^(-1),s≥-1/4.The local wellposedness is obtained by the transformation from the problem into a nonlinear Schrodinger type equation system and the contraction mapping theorem in a suitably modified Bourgain type space inspired by the work of Kishimoto,Tsugawa.This result improves the known local wellposedness in H^(s)×H^(s)×H^(s-1),s>-1/4 given by Farah.展开更多
In this paper we prove that the Cauchy problem associated with the generalized KdV-BO equation ut ＋ uxxx ＋ λH（uxx） ＋ u^2ux = 0, x ∈ R, t ≥ 0 is locally wellposed in Hr^s（R） for 4/3 〈r≤2, b〉1/r and s≥s（...In this paper we prove that the Cauchy problem associated with the generalized KdV-BO equation ut ＋ uxxx ＋ λH（uxx） ＋ u^2ux = 0, x ∈ R, t ≥ 0 is locally wellposed in Hr^s（R） for 4/3 〈r≤2, b〉1/r and s≥s（r）= 1/2- 1/2r. In particular, for r = 2, we reobtain the result in [3].展开更多
A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banac...A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of the evolution system is proved under some boundedness and smoothness conditions on the coefficient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with a singular convolution kernel, which is a generalized form of the stationary Wigner equation. Our investigation considerably improves the understanding of the open problem concerning the well-posedness of the stationary Wigner equation with in ow boundary conditions.展开更多
基金Supported by NSFC(10871175,10931007,10901137)Zhejiang Provincial Natural Science Foundation of China(Z6100217)Program for New Century Excellent Talents in University
文摘The weUposedness problem for an anisotropic incompressible viscous fluid in R3, ro- tating around a vector B（t, x） ：= （b1 （t, x）, b2 （t, x）, b3 （t, x））, is studied. The global wellposedness in the homogeneous case （B = e3） with sufficiently fast rotation in the space B0,1/2 is proved. In the inhomogeneous case （B = B（t, xh））, the global existence and uniqueness of the solution in B0,1/2 are obtained, provided that the initial data are sufficient small compared to the horizontal viscosity. Furthermore, we obtain uniform local existence and uniqueness of the solution in the x same function space. We also obtain propagation of the regularity in B2,11/2 under the additional assumption that B depends only on one horizontal space variable.
基金supported by National Natural Science Foundation of China(Grant Nos.11631007,11471174 and 11471259)。
文摘We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity.The primary goal of this paper is to understand how multi-component equations,nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons.We establish the local well-posedness of the Cauchy problem in Besov spaces B^s/p,r with 1 p,r+∞,s>max{1+1/p,3/2}and Sobolev spaces H^s(R)with s>3/2,and the method is based on the estimates for transport equations and new invariant properties of the system.Furthermore,the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied.A blow-up criterion on solutions of the Cauchy problem is demonstrated.In addition,we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line,and the single-peaked solitons on the circle,which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.
文摘In this paper we prove that the Schrodinger-Boussinesq system with solution(u,v,(-∂xx)-^(2/1)vt)is locally wellposed in H^(s)×H^(s)×Hs^(-1),s≥-1/4.The local wellposedness is obtained by the transformation from the problem into a nonlinear Schrodinger type equation system and the contraction mapping theorem in a suitably modified Bourgain type space inspired by the work of Kishimoto,Tsugawa.This result improves the known local wellposedness in H^(s)×H^(s)×H^(s-1),s>-1/4 given by Farah.
基金the Natural Science Foundation of Zhejiang Province （No. Y6080388） the Science and Technology Research Foundation of Zhejiang Ocean University （Nos. X08M014 X08Z04）.
文摘In this paper we prove that the Cauchy problem associated with the generalized KdV-BO equation ut ＋ uxxx ＋ λH（uxx） ＋ u^2ux = 0, x ∈ R, t ≥ 0 is locally wellposed in Hr^s（R） for 4/3 〈r≤2, b〉1/r and s≥s（r）= 1/2- 1/2r. In particular, for r = 2, we reobtain the result in [3].
基金National Natural Science Foundation of China (Grant Nos. 1167103& 91630130, 91434201, 11421101).
文摘A non-local abstract Cauchy problem with a singular integral is studied, which is a closed system of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, the well-posedness of the evolution system is proved under some boundedness and smoothness conditions on the coefficient functions. Furthermore, an isomorphism is established to extend the result to a partial integro-differential equation with a singular convolution kernel, which is a generalized form of the stationary Wigner equation. Our investigation considerably improves the understanding of the open problem concerning the well-posedness of the stationary Wigner equation with in ow boundary conditions.
