In this paper,we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms.We establish the global existence of mild solutions to this system with small in...In this paper,we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms.We establish the global existence of mild solutions to this system with small initial data.In addition,we also obtain the Gevrey class regularity and the temporal decay rate of the solution.展开更多
In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the diver...In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.展开更多
In this paper we prove local well-posedness in critical Besov spaces for the full compressible MHD equations in R^N, N≥ 2, under the assumptions that the initialdensity is bounded away from zero. The proof relies on ...In this paper we prove local well-posedness in critical Besov spaces for the full compressible MHD equations in R^N, N≥ 2, under the assumptions that the initialdensity is bounded away from zero. The proof relies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term.展开更多
In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that steins from an eventual lack of smoothness in the solution, and then ob...In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that steins from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.展开更多
This article is concerned with the 3 D nonhomogeneous incompressible magnetohydrodynamics equations with a slip boundary conditions in bounded domain.We obtain weighted estimates of the velocity and magnetic field,and...This article is concerned with the 3 D nonhomogeneous incompressible magnetohydrodynamics equations with a slip boundary conditions in bounded domain.We obtain weighted estimates of the velocity and magnetic field,and address the issue of local existence and uniqueness of strong solutions with the weaker initial data which contains vacuum states.展开更多
In this paper, we mainly study the global L2 stability for large solutions to the MHD equations in three-dimensional bounded or unbounded domains. Under suitable conditions of the large solutions, it is shown that the...In this paper, we mainly study the global L2 stability for large solutions to the MHD equations in three-dimensional bounded or unbounded domains. Under suitable conditions of the large solutions, it is shown that the large solutions are stable. And we obtain the equivalent condition of this stability condition. Moreover, the global existence and the stability of two-dimensional MHD equations under three-dimensional perturbations are also established.展开更多
In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics(MHD) equations in R^3. Then we establish that the solutions with initial data belo...In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics(MHD) equations in R^3. Then we establish that the solutions with initial data belonging to H^m(R^3) ∩ L^1(R^3) have the following time decay rate:║▽~mu(x, t) ║~2+║ ▽~mb(x, t)║~ 2+ ║▽^(m+1)u(x, t)║~ 2+ ║▽^(m+1)b(x, t) ║~2≤ c(1 + t)^(-3/2-m)for large t, where m = 0, 1.展开更多
In this paper,we consider the 2-D MHD equations with magnetic resistivity but without dissipation on the torus.We prove that if the initial data is small in H4(T2),then the 2-D MHD equations are globally well-posed.To...In this paper,we consider the 2-D MHD equations with magnetic resistivity but without dissipation on the torus.We prove that if the initial data is small in H4(T2),then the 2-D MHD equations are globally well-posed.To our knowledge,this is the first global well-posedness result for this system.展开更多
We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitabl...We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are HSlder continuous near boundary provided that the scaled mixed Lx,t^p,q-norm of the velocity vector field with 3/p + 2/q 〈 2, 2 〈 q 〈 ∞ is sufficiently small near the boundary. Also, we will investigate that for this 3 2〈3 solution U ∈ Lx,t^p,q with 1 〈 3+p +2/q+≤3/2, 3 〈 p 〈 ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}(3/q+2/q- 1).展开更多
In this paper, we study the Cauchy problem of the 2D incompressible magnetohydrodynamic equations in Lei-Lin space. The global well-posedness of a strong solution in the Lei-Lin space χ^(-1)(R^(2)) with any initial d...In this paper, we study the Cauchy problem of the 2D incompressible magnetohydrodynamic equations in Lei-Lin space. The global well-posedness of a strong solution in the Lei-Lin space χ^(-1)(R^(2)) with any initial data in χ^(-1)(R^(2)) ∩ L^(2)(R^(2)) is established. Furthermore, the uniqueness of the strong solution in χ^(-1)(R^(2)) and the Leray-Hopf weak solution in L^(2)(R^(2)) is proved.展开更多
A partial Runge-Kutta Discontinuous Galerkin(RKDG)method which preserves the exactly divergence-free property of the magnetic field is proposed in this paper to solve the two-dimensional ideal compressible magnetohydr...A partial Runge-Kutta Discontinuous Galerkin(RKDG)method which preserves the exactly divergence-free property of the magnetic field is proposed in this paper to solve the two-dimensional ideal compressible magnetohydrodynamics(MHD)equations written in semi-Lagrangian formulation on moving quadrilateral meshes.In this method,the fluid part of the ideal MHD equations along with zcomponent of the magnetic induction equation is discretized by the RKDG method as our previous paper[47].The numerical magnetic field in the remaining two directions(i.e.,x and y)are constructed by using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD.Since the divergence of the magnetic field in 2D is independent of its z-direction component,an exactly divergence-free numerical magnetic field can be obtained by this treatment.We propose a new nodal solver to improve the calculation accuracy of velocities of the moving meshes.A limiter is presented for the numerical solution of the fluid part of the MHD equations and it can avoid calculating the complex eigen-system of the MHD equations.Some numerical examples are presented to demonstrate the accuracy,non-oscillatory property and preservation of the exactly divergence-free property of our method.展开更多
In this paper,we construct a two-dimensional third-order space-time conservation element and solution element(CESE)method and apply it to the magnetohydrodynamics(MHD)equations.This third-order CESE method preserves a...In this paper,we construct a two-dimensional third-order space-time conservation element and solution element(CESE)method and apply it to the magnetohydrodynamics(MHD)equations.This third-order CESE method preserves all the favorable attributes of the original second-order CESEmethod,such as:(i)flux conservation in space and time without using an approximated Riemann solver,(ii)genuine multi-dimensional algorithm without dimensional splitting,(iii)the use of the most compact mesh stencil,involving only the immediate neighboring cells surrounding the cell where the solution at a new time step is sought,and(iv)an explicit,unified space-time integration procedure without using a quadrature integration procedure.In order to verify the accuracy and efficiency of the scheme,several 2D MHD test problems are presented.The result of MHD smooth wave problem shows third-order convergence of the scheme.The results of the other MHD test problems show that the method can enhance the solution quality by comparing with the original second-order CESE scheme.展开更多
.The regularity for 3-D MHD equations is considered in this paper,it is proved that the solutions(v,B,p)are Holder continuous if the velocity field v∈L^(∞)(0,T;L^(3,∞)_(x)(R^(3))with local small condition r^(-3)|{x....The regularity for 3-D MHD equations is considered in this paper,it is proved that the solutions(v,B,p)are Holder continuous if the velocity field v∈L^(∞)(0,T;L^(3,∞)_(x)(R^(3))with local small condition r^(-3)|{x∈B_(r)(x_(0):|v(x,t_(0))|>εr^(-1)}|≤ε and the magnetic field B∈L^(∞)(0,T;VMO^(-1)(R^(3)).展开更多
In this paper, we consider the blow-up of smooth solutions to the 3D ideal MHD equations. Let (u, b) be a smooth solution in (0, T). It is proved that the solution (u, b) can be extended after t = T if . This is an i...In this paper, we consider the blow-up of smooth solutions to the 3D ideal MHD equations. Let (u, b) be a smooth solution in (0, T). It is proved that the solution (u, b) can be extended after t = T if . This is an improvement of the result given by Caflisch, Klapper, and Steele [3].展开更多
In this paper, we prove the global well-posedness of the 3-D magnetohydrodynamics(MHD) equations with partial diffusion in the periodic domain when the initial velocity is small and the initial magnetic field is close...In this paper, we prove the global well-posedness of the 3-D magnetohydrodynamics(MHD) equations with partial diffusion in the periodic domain when the initial velocity is small and the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition.展开更多
In this paper,we investigate the solvability,regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magnetohydrodynamic(MHD)equations in bounded domains.On the boundary,the veloci...In this paper,we investigate the solvability,regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magnetohydrodynamic(MHD)equations in bounded domains.On the boundary,the velocity field fulfills a Navier-slip condition,while the magnetic field satisfies the insulating condition.It is shown that the initial boundary value problem has a global weak solution for a general smooth domain.More importantly,for a flat domain,we establish the uniform local well-posedness of the strong solution with higher-order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD equation as the dissipations tend to zero.展开更多
In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞(0, T; L^n(Ω). We prove that if the velocity u belongs to the critical space L∞(0, T...In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞(0, T; L^n(Ω). We prove that if the velocity u belongs to the critical space L∞(0, T; L^n(Ω), the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.展开更多
In this note, we will give a new proof of the blow-up criterion of smooth solutions to the 3D incompressible magneto-hydrodynamic equations by a simple application of Gagliardo-Nirenberg' s inequality.
A well-balanced second order finite volume central scheme for the magnetohydrodynamic(MHD)equations with gravitational source term is developed in this paper.The scheme is an unstaggered central scheme that evolves th...A well-balanced second order finite volume central scheme for the magnetohydrodynamic(MHD)equations with gravitational source term is developed in this paper.The scheme is an unstaggered central scheme that evolves the numerical solution on a single grid and avoids solving Riemann problems at the cell interfaces using ghost staggered cells.A subtraction technique is used on the conservative variables with the support of a known steady state in order to manifest the well-balanced property of the scheme.The divergence-free constraint of themagnetic field is satisfied after applying the constrained transport method(CTM)for unstaggered central schemes at the end of each time-step by correcting the components of the magnetic field.The robustness of the proposed scheme is verified on a list of numerical test cases from the literature.展开更多
In this paper,we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion.We first establish the local well-posedness(existence,uniqueness and continuous dependence)with initial d...In this paper,we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion.We first establish the local well-posedness(existence,uniqueness and continuous dependence)with initial data(u_(0),b_(0))in critical Besov spaces B_(p,1)^(d/p+1)×B_(p,1)^(d/p)with 1≤p≤∞,and give a lifespan T of the solution which depends on the norm of the Littlewood–Paley decomposition(profile)of the initial data.Then,we prove the global existence in critical Besov spaces.In particular,the results of global existence also hold in Sobolev space C([0,∞);H~s(S~2))×(C([0,∞);H^(s-1)(S~2))∩L~2([0,∞);H~s(S~2)))with s>2,when the initial data satisfies∫_(S~2)b_(0)dx=0 and||u_(0)||B_(()∞,1~((S~2)))~1+||b_(0)||B_(()∞,1^(S~2))~0≤ε.It’s worth noting that our results imply some large and low regularity initial data for the global existence.展开更多
基金supported by the Opening Project of Guangdong Province Key Laboratory of Cyber-Physical System(20168030301008)supported by the National Natural Science Foundation of China(11126266)+4 种基金the Natural Science Foundation of Guangdong Province(2016A030313390)the Quality Engineering Project of Guangdong Province(SCAU-2021-69)the SCAU Fund for High-level University Buildingsupported by the National Key Research and Development Program of China(2020YFA0712500)the National Natural Science Foundation of China(11971496,12126609)。
文摘In this paper,we study the three-dimensional regularized MHD equations with fractional Laplacians in the dissipative and diffusive terms.We establish the global existence of mild solutions to this system with small initial data.In addition,we also obtain the Gevrey class regularity and the temporal decay rate of the solution.
基金supported by the NSFC Grant 11901555,12271499the Cyrus Tang Foundationsupported by the NSFC Grant 11871448 and 12126604.
文摘In this paper,we construct a high-order discontinuous Galerkin(DG)method which can preserve the positivity of the density and the pressure for the viscous and resistive magnetohydrodynamics(VRMHD).To control the divergence error in the magnetic field,both the local divergence-free basis and the Godunov source term would be employed for the multi-dimensional VRMHD.Rigorous theoretical analyses are presented for one-dimensional and multi-dimensional DG schemes,respectively,showing that the scheme can maintain the positivity-preserving(PP)property under some CFL conditions when combined with the strong-stability-preserving time discretization.Then,general frameworks are established to construct the PP limiter for arbitrary order of accuracy DG schemes.Numerical tests demonstrate the effectiveness of the proposed schemes.
文摘In this paper we prove local well-posedness in critical Besov spaces for the full compressible MHD equations in R^N, N≥ 2, under the assumptions that the initialdensity is bounded away from zero. The proof relies on uniform estimates for a mixed hyperbolic/parabolic linear system with a convection term.
基金Supported by NSFC (10976026)supported by the Fundamental Research Funds for the Central Universities (11QZR18)the Research Funds for high-level talents of Huaqiao University (12BS232)
文摘In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that steins from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.
基金This work was supported by Natural Science Foundation of China(11871412).
文摘This article is concerned with the 3 D nonhomogeneous incompressible magnetohydrodynamics equations with a slip boundary conditions in bounded domain.We obtain weighted estimates of the velocity and magnetic field,and address the issue of local existence and uniqueness of strong solutions with the weaker initial data which contains vacuum states.
基金supported by 973 Program(2011CB711100)supported by NSFC (11171229)
文摘In this paper, we mainly study the global L2 stability for large solutions to the MHD equations in three-dimensional bounded or unbounded domains. Under suitable conditions of the large solutions, it is shown that the large solutions are stable. And we obtain the equivalent condition of this stability condition. Moreover, the global existence and the stability of two-dimensional MHD equations under three-dimensional perturbations are also established.
基金Supported by NSFC(11271290)GSPT of Zhejiang Province(2014R424062)
文摘In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics(MHD) equations in R^3. Then we establish that the solutions with initial data belonging to H^m(R^3) ∩ L^1(R^3) have the following time decay rate:║▽~mu(x, t) ║~2+║ ▽~mb(x, t)║~ 2+ ║▽^(m+1)u(x, t)║~ 2+ ║▽^(m+1)b(x, t) ║~2≤ c(1 + t)^(-3/2-m)for large t, where m = 0, 1.
文摘In this paper,we consider the 2-D MHD equations with magnetic resistivity but without dissipation on the torus.We prove that if the initial data is small in H4(T2),then the 2-D MHD equations are globally well-posed.To our knowledge,this is the first global well-posedness result for this system.
基金partly supported by BK21 PLUS SNU Mathematical Sciences Division and Basic Science Research Program through the National Research Foundation of Korea(NRF)(NRF-2016R1D1A1B03930422)
文摘We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are HSlder continuous near boundary provided that the scaled mixed Lx,t^p,q-norm of the velocity vector field with 3/p + 2/q 〈 2, 2 〈 q 〈 ∞ is sufficiently small near the boundary. Also, we will investigate that for this 3 2〈3 solution U ∈ Lx,t^p,q with 1 〈 3+p +2/q+≤3/2, 3 〈 p 〈 ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}(3/q+2/q- 1).
基金the National Natural Science Foundation of China (No. 11471103)。
文摘In this paper, we study the Cauchy problem of the 2D incompressible magnetohydrodynamic equations in Lei-Lin space. The global well-posedness of a strong solution in the Lei-Lin space χ^(-1)(R^(2)) with any initial data in χ^(-1)(R^(2)) ∩ L^(2)(R^(2)) is established. Furthermore, the uniqueness of the strong solution in χ^(-1)(R^(2)) and the Leray-Hopf weak solution in L^(2)(R^(2)) is proved.
基金supported by National Natural Science Foundation of China(Nos.12071046,11671049,91330107,11571002 and 11702028)China Postdoctoral Science Foundation(No.2020TQ0013).
文摘A partial Runge-Kutta Discontinuous Galerkin(RKDG)method which preserves the exactly divergence-free property of the magnetic field is proposed in this paper to solve the two-dimensional ideal compressible magnetohydrodynamics(MHD)equations written in semi-Lagrangian formulation on moving quadrilateral meshes.In this method,the fluid part of the ideal MHD equations along with zcomponent of the magnetic induction equation is discretized by the RKDG method as our previous paper[47].The numerical magnetic field in the remaining two directions(i.e.,x and y)are constructed by using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal MHD.Since the divergence of the magnetic field in 2D is independent of its z-direction component,an exactly divergence-free numerical magnetic field can be obtained by this treatment.We propose a new nodal solver to improve the calculation accuracy of velocities of the moving meshes.A limiter is presented for the numerical solution of the fluid part of the MHD equations and it can avoid calculating the complex eigen-system of the MHD equations.Some numerical examples are presented to demonstrate the accuracy,non-oscillatory property and preservation of the exactly divergence-free property of our method.
基金supported by the National Natural Science Foundation of China(Grant Nos.42030204,41874202)Shenzhen Natural Science Fund(the Stable Support Plan Program GXWD20220817152453003)+1 种基金Shenzhen Key Laboratory Launching Project(No.ZDSYS20210702140800001)the Specialized Research Fund for State Key Laboratories.
文摘In this paper,we construct a two-dimensional third-order space-time conservation element and solution element(CESE)method and apply it to the magnetohydrodynamics(MHD)equations.This third-order CESE method preserves all the favorable attributes of the original second-order CESEmethod,such as:(i)flux conservation in space and time without using an approximated Riemann solver,(ii)genuine multi-dimensional algorithm without dimensional splitting,(iii)the use of the most compact mesh stencil,involving only the immediate neighboring cells surrounding the cell where the solution at a new time step is sought,and(iv)an explicit,unified space-time integration procedure without using a quadrature integration procedure.In order to verify the accuracy and efficiency of the scheme,several 2D MHD test problems are presented.The result of MHD smooth wave problem shows third-order convergence of the scheme.The results of the other MHD test problems show that the method can enhance the solution quality by comparing with the original second-order CESE scheme.
基金This work was supported partly by NSFC(Grants 11971113,11631011)。
文摘.The regularity for 3-D MHD equations is considered in this paper,it is proved that the solutions(v,B,p)are Holder continuous if the velocity field v∈L^(∞)(0,T;L^(3,∞)_(x)(R^(3))with local small condition r^(-3)|{x∈B_(r)(x_(0):|v(x,t_(0))|>εr^(-1)}|≤ε and the magnetic field B∈L^(∞)(0,T;VMO^(-1)(R^(3)).
文摘In this paper, we consider the blow-up of smooth solutions to the 3D ideal MHD equations. Let (u, b) be a smooth solution in (0, T). It is proved that the solution (u, b) can be extended after t = T if . This is an improvement of the result given by Caflisch, Klapper, and Steele [3].
基金supported by National Natural Science Foundation of China(Grant No.11425103)supported by the Postdoctoral Science Foundation of China(Grant No.2019TQ0006)。
文摘In this paper, we prove the global well-posedness of the 3-D magnetohydrodynamics(MHD) equations with partial diffusion in the periodic domain when the initial velocity is small and the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition.
基金supported by National Natural Science Foundation of China(Grant No.11771300)supported by National Natural Science Foundation of China(Grant No.11871412)+2 种基金National Science Foundation of Guangdong(Grant No.2020A1515010554)supported by Zheng Ge Ru Foundation,Hong Kong Research Grants Council Earmarked Research(Grant Nos.CUHK14302819,CUHK14300917 and CUHK14302917)Basic and Applied Basic Research Foundation of Guangdong Province(Grant No.2020B1515310002)。
文摘In this paper,we investigate the solvability,regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magnetohydrodynamic(MHD)equations in bounded domains.On the boundary,the velocity field fulfills a Navier-slip condition,while the magnetic field satisfies the insulating condition.It is shown that the initial boundary value problem has a global weak solution for a general smooth domain.More importantly,for a flat domain,we establish the uniform local well-posedness of the strong solution with higher-order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD equation as the dissipations tend to zero.
基金supported by NSF of China (Grant Nos. 10431060 & 10771177)
文摘In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞(0, T; L^n(Ω). We prove that if the velocity u belongs to the critical space L∞(0, T; L^n(Ω), the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.
文摘In this note, we will give a new proof of the blow-up criterion of smooth solutions to the 3D incompressible magneto-hydrodynamic equations by a simple application of Gagliardo-Nirenberg' s inequality.
基金the National Council for Scientific Research of Lebanon(CNRS-L)for granting a doctoral fellowship to Farah Kanbarfunding by theQualification Programof the Julius Maximilians University Wurzburg.
文摘A well-balanced second order finite volume central scheme for the magnetohydrodynamic(MHD)equations with gravitational source term is developed in this paper.The scheme is an unstaggered central scheme that evolves the numerical solution on a single grid and avoids solving Riemann problems at the cell interfaces using ghost staggered cells.A subtraction technique is used on the conservative variables with the support of a known steady state in order to manifest the well-balanced property of the scheme.The divergence-free constraint of themagnetic field is satisfied after applying the constrained transport method(CTM)for unstaggered central schemes at the end of each time-step by correcting the components of the magnetic field.The robustness of the proposed scheme is verified on a list of numerical test cases from the literature.
基金Supported by National Natural Science Foundation of China(Grant No.11671407 and 11701586)the Macao Science and Technology Development Fund(Grant No.0091/2018/A3)+1 种基金Guangdong Special Support Program(Grant No.8-2015)the key pro ject of NSF of Guangdong province(Grant No.2016A030311004)。
文摘In this paper,we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion.We first establish the local well-posedness(existence,uniqueness and continuous dependence)with initial data(u_(0),b_(0))in critical Besov spaces B_(p,1)^(d/p+1)×B_(p,1)^(d/p)with 1≤p≤∞,and give a lifespan T of the solution which depends on the norm of the Littlewood–Paley decomposition(profile)of the initial data.Then,we prove the global existence in critical Besov spaces.In particular,the results of global existence also hold in Sobolev space C([0,∞);H~s(S~2))×(C([0,∞);H^(s-1)(S~2))∩L~2([0,∞);H~s(S~2)))with s>2,when the initial data satisfies∫_(S~2)b_(0)dx=0 and||u_(0)||B_(()∞,1~((S~2)))~1+||b_(0)||B_(()∞,1^(S~2))~0≤ε.It’s worth noting that our results imply some large and low regularity initial data for the global existence.