Using known Ca-multiplier result, we give necessary and sufficient conditions for the second order delay equations：u″（t）=Au（t）＋Fut＋Gu′＋f（t）,t∈Rto have maximal regularity in HSlder continuous function spac...Using known Ca-multiplier result, we give necessary and sufficient conditions for the second order delay equations：u″（t）=Au（t）＋Fut＋Gu′＋f（t）,t∈Rto have maximal regularity in HSlder continuous function spaces C^α （R, X）, where X is a Banach space, A is a closed operator in X, F, G ∈L（C（[-r, 0], X）, X） are delay operators for some fixed r 〉 0.展开更多
If the second order problem u（t） ＋ Bu（t） ＋ Au（t） = f（t）, u（0） =u（0） = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining th...If the second order problem u（t） ＋ Bu（t） ＋ Au（t） = f（t）, u（0） =u（0） = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining the estimates of ‖λ（λ^2 ＋ λB ＋ A）^-1‖ and ‖B（λ^2 ＋ λB ＋ A）^-1‖ for λ∈ C with Reλ 〉 ω, where the constant ω≥ 0.展开更多
The authors establish operator-valued Fourier multiplier theorems on Triebel spaces on R^N, where the required smoothness of the multiplier functions depends on the dimension N and the indices of the Triebel spaces. T...The authors establish operator-valued Fourier multiplier theorems on Triebel spaces on R^N, where the required smoothness of the multiplier functions depends on the dimension N and the indices of the Triebel spaces. This is used to give a sufficient condition of the maximal regularity in the sense of Triebel spaces for vector-valued Cauchy problems with Dirichlet boundary conditions.展开更多
The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition. A linearized map in a closed convex set is defined. The image set is precompact, and thus a fixe...The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition. A linearized map in a closed convex set is defined. The image set is precompact, and thus a fixed point exists. A multi-scale expansion method is used to obtain the homogenized equation. This equation satisfies a similar growth condition.展开更多
This paper is dedicated to the global-in-time existence and uniqueness issues of solutions for the inhomogeneous incompressible Hall-magnetohydrodynamics(MHD)system with merely bounded density.In a three-dimensional c...This paper is dedicated to the global-in-time existence and uniqueness issues of solutions for the inhomogeneous incompressible Hall-magnetohydrodynamics(MHD)system with merely bounded density.In a three-dimensional case,assuming that the initial density is a small perturbation of a positive constant in the L∞norm,we prove global well-posedness for small initial velocity and magnetic fields in critical Besov spaces.Next,we consider the so-called 21/2D flows for the inhomogeneous Hall-MHD system(that is 3D flows independent of the vertical variable)and establish the global existence of strong solutions by only assuming that the initial magnetic field is small in critical spaces and the initial density is bounded and bounded away from zero.In particular,those solutions allow piecewise constant density with jumps so that a mixture of fluids can be considered.Compared with inhomogeneous incompressible Navier-Stokes equations,the new difficulties of proving these results come from the additional so-called Hall term,which endows the magnetic equation with a quasi-linear character.In order to overcome them,we reformulate the system by taking advantage of the curl form of the magnetic equation and develop some new maximal regularity estimates for the Stokes system with just bounded coefficients.展开更多
This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal...This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of spaces E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued Lp,γ spaces is proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp,γ regularity and R-positivity uniformly with respect to these parameters.展开更多
We establish operator-valued Fourier multiplier theorems on periodic Triebel spaces, where the required smoothness of the multipliers depends on the indices of the Triebel spaces. This is used to give a characterizati...We establish operator-valued Fourier multiplier theorems on periodic Triebel spaces, where the required smoothness of the multipliers depends on the indices of the Triebel spaces. This is used to give a characterization of the maximal regularity in the sense of Triebel spaces for Cauchy problems with periodic boundary conditions.展开更多
Let X,Y be UMD-spaces that have property (α), 1< p< ∞ and let M be anR-bounded subset in L(X, Y). It is shown that {T(M_k)_(k∈z): M_k, k(M_(k+l)-M_k) ∈M for k∈Z} is an R-bounded subset of L(L^p (0,2π; X), ...Let X,Y be UMD-spaces that have property (α), 1< p< ∞ and let M be anR-bounded subset in L(X, Y). It is shown that {T(M_k)_(k∈z): M_k, k(M_(k+l)-M_k) ∈M for k∈Z} is an R-bounded subset of L(L^p (0,2π; X), L^p(0,2π; Y)), where T(M_m)_(k∈zdenotes the L^p-multiplier given by the sequence (M_k)_(k∈z), This generalizes a resultof Venni [10]. The author uses this result to study the strongly L^p-well-posedness ofevolution equations with periodic boundary condition. Analogous results for operator-valued L^p-multipliers on R are also given.展开更多
In this paper, we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions. The proof is obtained by a bootstrap argument and maximal...In this paper, we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions. The proof is obtained by a bootstrap argument and maximal regularity estimates. If n≥ 10/3m, we also give another proof which does not use maximal regularity estimates.展开更多
The free boundary value problems for elliptic differential-operator equations are studied.Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L p-...The free boundary value problems for elliptic differential-operator equations are studied.Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L p-spaces are given.In application,the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.展开更多
This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in L_p spaces. These results are a...This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in L_p spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed L_p norm.展开更多
基金supported by the NSF of China (No. 10571099)Specialized Research Fund for the Doctoral Program of Higher Educationthe Tsinghua Basic Research Foundation (JCpy2005056)
文摘Using known Ca-multiplier result, we give necessary and sufficient conditions for the second order delay equations：u″（t）=Au（t）＋Fut＋Gu′＋f（t）,t∈Rto have maximal regularity in HSlder continuous function spaces C^α （R, X）, where X is a Banach space, A is a closed operator in X, F, G ∈L（C（[-r, 0], X）, X） are delay operators for some fixed r 〉 0.
基金Supported by National Natural Science Foundation of China (Grant No. 10672062)
文摘If the second order problem u（t） ＋ Bu（t） ＋ Au（t） = f（t）, u（0） =u（0） = 0 has L^p-maximal regularity, 1 〈 p 〈 ∞, the analyticity of the corresponding propagator of the sine type is shown by obtaining the estimates of ‖λ（λ^2 ＋ λB ＋ A）^-1‖ and ‖B（λ^2 ＋ λB ＋ A）^-1‖ for λ∈ C with Reλ 〉 ω, where the constant ω≥ 0.
文摘The authors establish operator-valued Fourier multiplier theorems on Triebel spaces on R^N, where the required smoothness of the multiplier functions depends on the dimension N and the indices of the Triebel spaces. This is used to give a sufficient condition of the maximal regularity in the sense of Triebel spaces for vector-valued Cauchy problems with Dirichlet boundary conditions.
基金Supported by the National Basic Research Program of China(973 Program)(No.2012CB025904)the National Natural Science Foundation of China(No.90916027)
文摘The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition. A linearized map in a closed convex set is defined. The image set is precompact, and thus a fixed point exists. A multi-scale expansion method is used to obtain the homogenized equation. This equation satisfies a similar growth condition.
基金supported by the Agence Nationale de la Recherche Project BORDS(Grant No.ANR-16-CE40-0027-01)the Labex MME-DII+2 种基金and the CY Initiative of ExcellenceProject CYNA(CY Nonlinear Analysis)supported by National Natural Science Foundation of China(Grant No.12101472)。
文摘This paper is dedicated to the global-in-time existence and uniqueness issues of solutions for the inhomogeneous incompressible Hall-magnetohydrodynamics(MHD)system with merely bounded density.In a three-dimensional case,assuming that the initial density is a small perturbation of a positive constant in the L∞norm,we prove global well-posedness for small initial velocity and magnetic fields in critical Besov spaces.Next,we consider the so-called 21/2D flows for the inhomogeneous Hall-MHD system(that is 3D flows independent of the vertical variable)and establish the global existence of strong solutions by only assuming that the initial magnetic field is small in critical spaces and the initial density is bounded and bounded away from zero.In particular,those solutions allow piecewise constant density with jumps so that a mixture of fluids can be considered.Compared with inhomogeneous incompressible Navier-Stokes equations,the new difficulties of proving these results come from the additional so-called Hall term,which endows the magnetic equation with a quasi-linear character.In order to overcome them,we reformulate the system by taking advantage of the curl form of the magnetic equation and develop some new maximal regularity estimates for the Stokes system with just bounded coefficients.
基金This work is supported by the grant of Istanbul University (Project UDP-227/18022004)
文摘This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of spaces E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued Lp,γ spaces is proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp,γ regularity and R-positivity uniformly with respect to these parameters.
基金The first author is supported by the NSF of China the Excellent Young Teachers Program of MOE,P.R.C.
文摘We establish operator-valued Fourier multiplier theorems on periodic Triebel spaces, where the required smoothness of the multipliers depends on the indices of the Triebel spaces. This is used to give a characterization of the maximal regularity in the sense of Triebel spaces for Cauchy problems with periodic boundary conditions.
基金Project supported by the National Natural Science Foundation of China (No.10271064) and the Excel-lent Young Teachers Program of the Ministry of Education of China
文摘Let X,Y be UMD-spaces that have property (α), 1< p< ∞ and let M be anR-bounded subset in L(X, Y). It is shown that {T(M_k)_(k∈z): M_k, k(M_(k+l)-M_k) ∈M for k∈Z} is an R-bounded subset of L(L^p (0,2π; X), L^p(0,2π; Y)), where T(M_m)_(k∈zdenotes the L^p-multiplier given by the sequence (M_k)_(k∈z), This generalizes a resultof Venni [10]. The author uses this result to study the strongly L^p-well-posedness ofevolution equations with periodic boundary condition. Analogous results for operator-valued L^p-multipliers on R are also given.
文摘In this paper, we derive a priori bounds for global solutions of 2m-th order semilinear parabolic equations with superlinear and subcritical growth conditions. The proof is obtained by a bootstrap argument and maximal regularity estimates. If n≥ 10/3m, we also give another proof which does not use maximal regularity estimates.
文摘The free boundary value problems for elliptic differential-operator equations are studied.Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L p-spaces are given.In application,the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.
文摘This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in L_p spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed L_p norm.
