This paper is concerned with the Cauchy problem for a 3D fluid-particle interaction model in the so-called flowing regime inℝ3.Under the smallness assumption on both the external potential and the initial perturbation...This paper is concerned with the Cauchy problem for a 3D fluid-particle interaction model in the so-called flowing regime inℝ3.Under the smallness assumption on both the external potential and the initial perturbation of the stationary solution in some Sobolev spaces,the existence and uniqueness of global smooth solutions in H3 of the system are established by using the careful energy method.展开更多
In this article, we consider the blowup criterion for the local strong solution to the compressible fluid-particle interaction model in dimension three with vacuum. We establish a BKM type criterion for possible break...In this article, we consider the blowup criterion for the local strong solution to the compressible fluid-particle interaction model in dimension three with vacuum. We establish a BKM type criterion for possible breakdown of such solutions at critical time in terms of both the L^∞ （0, T; L^6）-norm of the density of particles and the ^L1（0, T; L^∞）-norm of the deformation tensor of velocity gradient.展开更多
This paper addresses the 3-D Cauchy problem of a fluid-particle system with a magnetic field.First,the local classical solutions of the linearized model on the sphere Br are obtained by some a priori estimates that do...This paper addresses the 3-D Cauchy problem of a fluid-particle system with a magnetic field.First,the local classical solutions of the linearized model on the sphere Br are obtained by some a priori estimates that do not depend on the radius r.Second,the classical solutions of the linearized model in R^(3) are obtained by combining the continuation and compactness methods.Finally,the classical solutions of the original system are proved by use of the picard iteration argument and the energy method.展开更多
Main mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence are presented and discussed. The underlying fact is that there does not exist the absolute zero fluid-volume. T...Main mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence are presented and discussed. The underlying fact is that there does not exist the absolute zero fluid-volume. Therefore, the physical object corresponding to the absolute point is just the uniform fluid-particle. The fluid-particle, in general, corresponds to the monad. The uniform fluid-particle corresponds to the uniform monad, while the nonuniform fluid-particle to the nonuniform monad. There are two kinds of the differentiations, one is based on the absolute point, and the other based on the monad. The former is adopted in the Navier-Stokes equations, and the latter in the fundamental equations presented in this paper for the nonstandard analysis theory of turbulence. The continuity of fluid is elucidated by virtue of the concepts of the fluid-particle and fluid-particle at a lower level. Furthermore, the characters of the continuity in two cases, i.e. in the standard and nonstandard analyses, are presented in this paper. And the difference in discretization between the Navier-Stokes equations and the fundamental equations given herein is also pointed out.展开更多
文摘This paper is concerned with the Cauchy problem for a 3D fluid-particle interaction model in the so-called flowing regime inℝ3.Under the smallness assumption on both the external potential and the initial perturbation of the stationary solution in some Sobolev spaces,the existence and uniqueness of global smooth solutions in H3 of the system are established by using the careful energy method.
基金supported by the National Basic Research Program of China(973 Program)(2011CB808002)the National Natural Science Foundation of China(11371152,11128102,11071086,and 11571117)+3 种基金the Natural Science Foundation of Guangdong Province(S2012010010408)the Foundation for Distinguished Young Talents in Higher Education of Guangdong(2015KQNCX095)the Major Foundation of Hanshan Normal University(LZ201403)the Scientific Research Foundation of Graduate School of South China Normal University(2014ssxm04)
文摘In this article, we consider the blowup criterion for the local strong solution to the compressible fluid-particle interaction model in dimension three with vacuum. We establish a BKM type criterion for possible breakdown of such solutions at critical time in terms of both the L^∞ （0, T; L^6）-norm of the density of particles and the ^L1（0, T; L^∞）-norm of the deformation tensor of velocity gradient.
基金supported by the National Natural Science Foundation of China(12026253,12026244,11971357)the Natural Science Foundation of Guangdong Province(2018A030310008,2021A1515010303)+6 种基金Guangdong Key Laboratory for Functional Substances in Medicinal Edible Resources and Healthcare Products(2021B1212040015)NSF of Guangdong Provincial Department of Education(2019KTSCX097)Chaozhou Science and Technology plan project(2019ZC02)supported by the Key Project of National Natural Science Foundation of China(12131010)the National Natural Science Foundation of China(11771155,11571117,11871005)the Natural Science Foundation of Guangdong Province(2017A030313003,2019A1515011491,2021A1515010249)the Science and Technology Program of Guangzhou(2019050001)。
文摘This paper addresses the 3-D Cauchy problem of a fluid-particle system with a magnetic field.First,the local classical solutions of the linearized model on the sphere Br are obtained by some a priori estimates that do not depend on the radius r.Second,the classical solutions of the linearized model in R^(3) are obtained by combining the continuation and compactness methods.Finally,the classical solutions of the original system are proved by use of the picard iteration argument and the energy method.
基金Project supported by the National Natural Science Foundation of China （Grant No 10572135）.
文摘Main mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence are presented and discussed. The underlying fact is that there does not exist the absolute zero fluid-volume. Therefore, the physical object corresponding to the absolute point is just the uniform fluid-particle. The fluid-particle, in general, corresponds to the monad. The uniform fluid-particle corresponds to the uniform monad, while the nonuniform fluid-particle to the nonuniform monad. There are two kinds of the differentiations, one is based on the absolute point, and the other based on the monad. The former is adopted in the Navier-Stokes equations, and the latter in the fundamental equations presented in this paper for the nonstandard analysis theory of turbulence. The continuity of fluid is elucidated by virtue of the concepts of the fluid-particle and fluid-particle at a lower level. Furthermore, the characters of the continuity in two cases, i.e. in the standard and nonstandard analyses, are presented in this paper. And the difference in discretization between the Navier-Stokes equations and the fundamental equations given herein is also pointed out.
