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SOME DISCUSSIONS ON wellposedness OF THE EULER EQUATION

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Annals of Differential Equations 2007年第2期23卷 127-130页

作者： Chen Min Dept. of Math. Shanghai University Shanghai 200444

In this paper a new formulation for Euler equation is presented, the wellposedness for the new equation in Sobolev spaces is discussed.

关键词： Euler equations wellposedness

Well-posedness of a non-local abstract Cauchy problem with a singular integral

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Frontiers of Mathematics in China 2019年第1期14卷 77-93页

作者： Haiyan JIANG Tiao LU Xiangjiang ZHU Introduction We consider the following initial value problem for the unknown functions c(t) and z): 1 School of Mathematical Sciences Beijing Institute of Technology Beijing 100081 China School of Mathematical Sciences Peking University Beijing 100871 China CAPT HEDPS LMAM IFSA Collaborative Innovation Center of MoE Peking University Beijing 100871 China

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.

关键词： Partial integro-differential equation (PIDE) singular integral wellposedness Wigner equation

The Cauchy Problem of a Shallow Water Equation

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Acta Mathematica Sinica,English Series 2005年第2期21卷 393-408页

作者： XiaoFengLIU YongYangJIN DepartmentofMathematics EastChinaNormalUniversityShanghai200062P.R.China DepartmentofApplicationMathematics ZhejiangUniversityofTechnologyHangzhou310032P.R.China

We consider the Cauchy problem of a shallow water equation and its localwellposedness.

关键词： cauchy problem wellposedness shallow water equation

On the Cauchy problem and peakons of a two-component Novikov system

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Science China Mathematics 2020年第10期63卷 1965-1996页

作者： Changzheng Qu Ying Fu School of Mathematics and Statistics Ningbo UniversityNingbo 315211China School of Mathematics Northwest UniversityXi’an 710127China

We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic *** 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 *** 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 ***,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 *** 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.

关键词： two-component Novikov system Hamiltonian structure Camassa-Holm type equation wellposedness peaked soliton

Well-Posedness and Regularity of Partial Differential Equation Control Systems

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Well-Posedness and Regularity of Partial Differential Equati...