Let(u,B)be a strong solution of the magneto-hydrodynamic system on three dimensional torus T^(3).In this note,using the properties of the curl operator,we show that‖(▽×(u-B),▽×(u+B))(·,t)‖L^(1)+1/2v...Let(u,B)be a strong solution of the magneto-hydrodynamic system on three dimensional torus T^(3).In this note,using the properties of the curl operator,we show that‖(▽×(u-B),▽×(u+B))(·,t)‖L^(1)+1/2v‖(u-B,u+B)(·,t)‖_(L^(2))^(2)is decreasing in time t as long as the solution(u,B)(·,t)exists,where∇×w means the curl of the vector function w,and v>0 is the viscosity coefficient.展开更多
When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to ...When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to accomplish this, but variations on this theme exist, and we examine a few such variations. The mathematical analysis of is sought in the form if such an inverse operator exists, but physics is defined by both mathematical formula and ontological formalism, as I show for an example based on the Dirac equation. Finally, I contrast these “standard” approaches with a novel exact inverse operator for field equations.展开更多
This paper concerns the minimization problem of L2 norm of curl of vector fields prescribed full trace on the boundary of a multiconnected bounded domain.The existence of the minimizers in H1 are shown by orthogonal d...This paper concerns the minimization problem of L2 norm of curl of vector fields prescribed full trace on the boundary of a multiconnected bounded domain.The existence of the minimizers in H1 are shown by orthogonal decompositions of vector function spaces and a constructed auxiliary variational problem.And the H2 estimate of the type II divergence-free part of the minimizers is established by div-curl-gradient type estimates of vector fields.展开更多
基金supported by the National Natural Science Foundation of China(12371123).
文摘Let(u,B)be a strong solution of the magneto-hydrodynamic system on three dimensional torus T^(3).In this note,using the properties of the curl operator,we show that‖(▽×(u-B),▽×(u+B))(·,t)‖L^(1)+1/2v‖(u-B,u+B)(·,t)‖_(L^(2))^(2)is decreasing in time t as long as the solution(u,B)(·,t)exists,where∇×w means the curl of the vector function w,and v>0 is the viscosity coefficient.
文摘When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to accomplish this, but variations on this theme exist, and we examine a few such variations. The mathematical analysis of is sought in the form if such an inverse operator exists, but physics is defined by both mathematical formula and ontological formalism, as I show for an example based on the Dirac equation. Finally, I contrast these “standard” approaches with a novel exact inverse operator for field equations.
基金Supported by the National Natural Science Foundation of China(11501109)Designated Scientific Research Project of Provincial Universities of Fujian Province(JK2015014)。
文摘This paper concerns the minimization problem of L2 norm of curl of vector fields prescribed full trace on the boundary of a multiconnected bounded domain.The existence of the minimizers in H1 are shown by orthogonal decompositions of vector function spaces and a constructed auxiliary variational problem.And the H2 estimate of the type II divergence-free part of the minimizers is established by div-curl-gradient type estimates of vector fields.
