In this paper, we are going to consider the initial value problem with periodic boundarycondition and the Cauchy problem associated with the system of ferromagnetic chain withthe Gilbert damping term Z_t=-εZ×(Z&...In this paper, we are going to consider the initial value problem with periodic boundarycondition and the Cauchy problem associated with the system of ferromagnetic chain withthe Gilbert damping term Z_t=-εZ×(Z×Z_(zz)+Z×Z_(zz), (ε≥0).The existence of a unique smooth solution is established by using the technique of spatialdifference and crucial a priori estimates of higher-order derivatives in Sobolev spaces.展开更多
It is proved that any weak solution to the initial value problem of two dimensional Landau-Lifshitz equation is unique and is smooth with the exception of at most finitely many points, provided that the weak solution ...It is proved that any weak solution to the initial value problem of two dimensional Landau-Lifshitz equation is unique and is smooth with the exception of at most finitely many points, provided that the weak solution has finite energy.展开更多
We prove the existence of solutions of the static Landau-Lifshitz equation with multi- direct effective field and with Dirichlet boundary condition,and establish the stability of the solution of Landau-Lifshitz equati...We prove the existence of solutions of the static Landau-Lifshitz equation with multi- direct effective field and with Dirichlet boundary condition,and establish the stability of the solution of Landau-Lifshitz equation with respect to time.展开更多
The existence of global weak (smooth) solutions to the generalized Landau-Lifshitz systems of the ferromagnetic spin chain type from a Riemarm surface onto a unit sphere is established and some relation between harmon...The existence of global weak (smooth) solutions to the generalized Landau-Lifshitz systems of the ferromagnetic spin chain type from a Riemarm surface onto a unit sphere is established and some relation between harmonic maps and the solutions of the generalized Landau-Lifshitz system is found.展开更多
In this paper, we shall construct some explicit piecewise smooth （global continuous） solutions as well as blow up solutions to the multidimensional Landau-Lifshitz equation, subject to the external magnetic fields b...In this paper, we shall construct some explicit piecewise smooth （global continuous） solutions as well as blow up solutions to the multidimensional Landau-Lifshitz equation, subject to the external magnetic fields being both discontinuous and unbounded. When the external magnetic field is continuous, some explicit exact smooth solutions and blow up solution are also constructed. We also establish some necessary and sufficient conditions to ensure that the solution of multidimensional Landau-Lifshitz equation with external magnetic field converges to the solution of equation without external magnetic field when the external magnetic field tends to zero.展开更多
In this paper,we study the Cauchy problem of an integrable evolution system,i.e.,the n-dimensional generalization of third-order symmetry of the well-known Landau-Lifshitz equation.By rewriting this equation in a geom...In this paper,we study the Cauchy problem of an integrable evolution system,i.e.,the n-dimensional generalization of third-order symmetry of the well-known Landau-Lifshitz equation.By rewriting this equation in a geometric form and applying the geometric energy method with a forth-order perturbation,we show the global well-posedness of the Cauchy problem in suitable Sobolev spaces.展开更多
In this paper, we establish the global existence of the solution for the Landau-Lifshitz equation of the ferromagnetic spin chain. By Galerkin method, we first show the existence of the local solution for this equatio...In this paper, we establish the global existence of the solution for the Landau-Lifshitz equation of the ferromagnetic spin chain. By Galerkin method, we first show the existence of the local solution for this equation, and then by a priori estimates method, we extend the local solution to a global solution.展开更多
In this manuscript,we study a new version of the optical recursional binormal microbeam model for a flexible binormal microscale beam in terms of a binormal normalized operator.Also,we give new explanations for the op...In this manuscript,we study a new version of the optical recursional binormal microbeam model for a flexible binormal microscale beam in terms of a binormal normalized operator.Also,we give new explanations for the optical recursional visco Landau-Lifshitz binormal electromagnetic binormal microscale beam.Finally,we obtain an optical application for the normalized visco Landau-Lifshitz electromagnetic binormal optimistic density with an optical binormal resonator.展开更多
In this paper, we describe several stationary conditions on weak solutions to the inhomogeneous Landau-Lifshitz equation, which ensure the partial regularity. For certain class of proper stationary weak solutions, a c...In this paper, we describe several stationary conditions on weak solutions to the inhomogeneous Landau-Lifshitz equation, which ensure the partial regularity. For certain class of proper stationary weak solutions, a compactness result of the solutions, a finite Hausdorff measure result of the t-slice energy concentration sets and an asymptotic limit result of the Radon measures are proved. We also present a subtle rectifiability result for the energy concentration set of certain sequence of strong stationary weak solutions.展开更多
In this note, we prove the partial regularity of stationary weak solutions for the Landau- Lifshitz system with general potential in four or three dimensional space. As well known, in general, since the constraint of ...In this note, we prove the partial regularity of stationary weak solutions for the Landau- Lifshitz system with general potential in four or three dimensional space. As well known, in general, since the constraint of the methods, in order to get the partial regularity of stationary weak solution of the Landau-Lifshitz system with potential, we need to add some very strongly conditions on the potential. The main difficulty caused by potential is how to find the equation satisfied by the scaling function, which breaks down the blow-up processing. We estimate directly Morrey's energy to avoid the difficulties by blowing up.展开更多
We consider the regularity problem under the critical condition to some liquid crystal models and the Landau-Lifshitz equations. The Serrin type reularity criteria are obtained in the terms of the Besov spaces.
In this paper, based on classical Lie group method, we study multi-dimensional Landau-Lifshitz equation, and get its infinitesimal generator, symmetry group and new solutions. In particular, we build the connection be...In this paper, based on classical Lie group method, we study multi-dimensional Landau-Lifshitz equation, and get its infinitesimal generator, symmetry group and new solutions. In particular, we build the connection between new exact solutions and old exact solutions. At the same time, we also prove that the initial boundary value condition of the three-dimensional Landau-Lifshitz equation admits a unique solution and discuss the stability of the solution.展开更多
A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equation...A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equations. Then the Lie group method and the Runge-Kutta (RK) method were applied to the ordinary differential equations. The square conserving property and the accuracy of the two methods were compared. Numerical experiment results show the Lie group method has the good accuracy and the square conserving property than the RK method.展开更多
In this paper, we propose a model in studying soft ferromagnetic films, which is readily accessible experimentally. By using penalty approximation and compensated compactness, we prove that the dynamical equation in t...In this paper, we propose a model in studying soft ferromagnetic films, which is readily accessible experimentally. By using penalty approximation and compensated compactness, we prove that the dynamical equation in thin film has a local weak solution. Moreover, the corresponding linear equation is also dealt with in great detail.展开更多
We establish an extension result of existence and partial regularity for the nonzero Neumann initial-boundary value problem of the Landau-Lifshitz equation with nonpositive anisotropy constants in three or four dimens...We establish an extension result of existence and partial regularity for the nonzero Neumann initial-boundary value problem of the Landau-Lifshitz equation with nonpositive anisotropy constants in three or four dimension space. The partial regularity is proved up to the boundary and this result is an important supplement to those for the Dirichlet problem or the homogeneous Neumann problem.展开更多
In this paper, we are concerned with the existence and uniqueness of global smooth solu- tion for the Robin boundary value problem of Landau Lifshitz equations in one dimension when the boundary value depends on time ...In this paper, we are concerned with the existence and uniqueness of global smooth solu- tion for the Robin boundary value problem of Landau Lifshitz equations in one dimension when the boundary value depends on time t. Yhrthermore, by viscosity vanishing approach, we get the existence and uniqueness of the problem without Gilbert damping term when the boundary value is independent of t.展开更多
We consider the partial regularity of weak solutions to the weighted Landau-Lifshitz flow on a 2-dimensional bounded smooth domain by Ginzburg-Landau type approximation. Under the energy smallness condition, we prove ...We consider the partial regularity of weak solutions to the weighted Landau-Lifshitz flow on a 2-dimensional bounded smooth domain by Ginzburg-Landau type approximation. Under the energy smallness condition, we prove the uniform local C^∞ bounds for the approaching solutions. This shows that the approximating solutions are locally uniformly bounded in C^∞（Reg（{uε}）∩（Ω^-×R^＋）） which guarantee the smooth convergence in these points. Energy estimates for the approximating equations are used to prove that the singularity set has locally finite two-dimensional parabolic Hausdorff measure and has at most finite points at each fixed time. From the uniform boundedness of approximating solutions in C^∞（Reg（{uε}）∩（Ω^-×R^＋））, we then extract a subsequence converging to a global weak solution to the weighted Landau-Lifshitz flow which is in fact regular away from finitely many points.展开更多
基金Project supported by the National Natural Science Foundation of China.
文摘In this paper, we are going to consider the initial value problem with periodic boundarycondition and the Cauchy problem associated with the system of ferromagnetic chain withthe Gilbert damping term Z_t=-εZ×(Z×Z_(zz)+Z×Z_(zz), (ε≥0).The existence of a unique smooth solution is established by using the technique of spatialdifference and crucial a priori estimates of higher-order derivatives in Sobolev spaces.
文摘It is proved that any weak solution to the initial value problem of two dimensional Landau-Lifshitz equation is unique and is smooth with the exception of at most finitely many points, provided that the weak solution has finite energy.
文摘We prove the existence of solutions of the static Landau-Lifshitz equation with multi- direct effective field and with Dirichlet boundary condition,and establish the stability of the solution of Landau-Lifshitz equation with respect to time.
文摘The existence of global weak (smooth) solutions to the generalized Landau-Lifshitz systems of the ferromagnetic spin chain type from a Riemarm surface onto a unit sphere is established and some relation between harmonic maps and the solutions of the generalized Landau-Lifshitz system is found.
基金Supported by the National Natural Science Foundation of China(No.10861014)The Institute of Mathematical Sciences,Chinese University of Hongkong.
文摘In this paper, we shall construct some explicit piecewise smooth （global continuous） solutions as well as blow up solutions to the multidimensional Landau-Lifshitz equation, subject to the external magnetic fields being both discontinuous and unbounded. When the external magnetic field is continuous, some explicit exact smooth solutions and blow up solution are also constructed. We also establish some necessary and sufficient conditions to ensure that the solution of multidimensional Landau-Lifshitz equation with external magnetic field converges to the solution of equation without external magnetic field when the external magnetic field tends to zero.
基金supported by National Basic Research Program of China(Grant No.2006CB805902)
文摘In this paper,we study the Cauchy problem of an integrable evolution system,i.e.,the n-dimensional generalization of third-order symmetry of the well-known Landau-Lifshitz equation.By rewriting this equation in a geometric form and applying the geometric energy method with a forth-order perturbation,we show the global well-posedness of the Cauchy problem in suitable Sobolev spaces.
文摘In this paper, we establish the global existence of the solution for the Landau-Lifshitz equation of the ferromagnetic spin chain. By Galerkin method, we first show the existence of the local solution for this equation, and then by a priori estimates method, we extend the local solution to a global solution.
文摘In this manuscript,we study a new version of the optical recursional binormal microbeam model for a flexible binormal microscale beam in terms of a binormal normalized operator.Also,we give new explanations for the optical recursional visco Landau-Lifshitz binormal electromagnetic binormal microscale beam.Finally,we obtain an optical application for the normalized visco Landau-Lifshitz electromagnetic binormal optimistic density with an optical binormal resonator.
基金Project supported by the National Natural Science Foundation of China (No. 10571158)the Natural Science Foundation of Zheji-ang Province, China (No. Y605076)
文摘In this paper, we describe several stationary conditions on weak solutions to the inhomogeneous Landau-Lifshitz equation, which ensure the partial regularity. For certain class of proper stationary weak solutions, a compactness result of the solutions, a finite Hausdorff measure result of the t-slice energy concentration sets and an asymptotic limit result of the Radon measures are proved. We also present a subtle rectifiability result for the energy concentration set of certain sequence of strong stationary weak solutions.
基金supported by National Natural Science Foundation of China (Grant Nos. 10631020, 60850005)the Natural Science Foundation of Zhejiang Province (Grant No. D7080080)
文摘In this note, we prove the partial regularity of stationary weak solutions for the Landau- Lifshitz system with general potential in four or three dimensional space. As well known, in general, since the constraint of the methods, in order to get the partial regularity of stationary weak solution of the Landau-Lifshitz system with potential, we need to add some very strongly conditions on the potential. The main difficulty caused by potential is how to find the equation satisfied by the scaling function, which breaks down the blow-up processing. We estimate directly Morrey's energy to avoid the difficulties by blowing up.
基金the National Natural Science Foundation of China (Grant No. 10301014)
文摘We consider the regularity problem under the critical condition to some liquid crystal models and the Landau-Lifshitz equations. The Serrin type reularity criteria are obtained in the terms of the Besov spaces.
文摘In this paper, based on classical Lie group method, we study multi-dimensional Landau-Lifshitz equation, and get its infinitesimal generator, symmetry group and new solutions. In particular, we build the connection between new exact solutions and old exact solutions. At the same time, we also prove that the initial boundary value condition of the three-dimensional Landau-Lifshitz equation admits a unique solution and discuss the stability of the solution.
文摘A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equations. Then the Lie group method and the Runge-Kutta (RK) method were applied to the ordinary differential equations. The square conserving property and the accuracy of the two methods were compared. Numerical experiment results show the Lie group method has the good accuracy and the square conserving property than the RK method.
基金the National Natural Science Foundation of China(No.10171113,10471156)(Tianyuan Foundation 10526040)Guangdong Provincial Natural Science Foundation(No.4009793)
文摘In this paper, we propose a model in studying soft ferromagnetic films, which is readily accessible experimentally. By using penalty approximation and compensated compactness, we prove that the dynamical equation in thin film has a local weak solution. Moreover, the corresponding linear equation is also dealt with in great detail.
基金Supported by the Science Foundation of Zhejiang Sci-Tech University(No.0905828-Y)
文摘We establish an extension result of existence and partial regularity for the nonzero Neumann initial-boundary value problem of the Landau-Lifshitz equation with nonpositive anisotropy constants in three or four dimension space. The partial regularity is proved up to the boundary and this result is an important supplement to those for the Dirichlet problem or the homogeneous Neumann problem.
基金Supported by National Basic Research Program of China (973 Program) (Grant No. 2011CB808002)National Natural Science Foundation of China (Grant Nos. 11071086 and 11128102)the University Special Re-search Foundation for Ph.D. Program (Grant No. 20104407110002)
文摘In this paper, we are concerned with the existence and uniqueness of global smooth solu- tion for the Robin boundary value problem of Landau Lifshitz equations in one dimension when the boundary value depends on time t. Yhrthermore, by viscosity vanishing approach, we get the existence and uniqueness of the problem without Gilbert damping term when the boundary value is independent of t.
基金The second author is partially supported by the National Natural Science Foundation of China （Grant No.10471050）, the National 973 project （Grant No. 2006CB805902） and by Guangdong Provincial Natural Science Foundation （Grant No.031495）.
文摘We consider the partial regularity of weak solutions to the weighted Landau-Lifshitz flow on a 2-dimensional bounded smooth domain by Ginzburg-Landau type approximation. Under the energy smallness condition, we prove the uniform local C^∞ bounds for the approaching solutions. This shows that the approximating solutions are locally uniformly bounded in C^∞（Reg（{uε}）∩（Ω^-×R^＋）） which guarantee the smooth convergence in these points. Energy estimates for the approximating equations are used to prove that the singularity set has locally finite two-dimensional parabolic Hausdorff measure and has at most finite points at each fixed time. From the uniform boundedness of approximating solutions in C^∞（Reg（{uε}）∩（Ω^-×R^＋））, we then extract a subsequence converging to a global weak solution to the weighted Landau-Lifshitz flow which is in fact regular away from finitely many points.
