This paper deals with the Gevreg-hypoellipticity for a class of totally characteristic operators with the elliptic condition and the discrete boundary spectrum condition respectively.
In this paper,we use the method of representation of Lie group to study a class of nonhomoge- neous convolution operator on the nilpotent Lie group H^M×R^k,and give a criteerion of their hypoellipticity.
Let G be a connected and simply-connected step two homogeneous group, K be a left invariant homogeneous convolution operator on G. If K is hypoelliptic, then (K) is injective on C for every nontrivial irreducible unit...Let G be a connected and simply-connected step two homogeneous group, K be a left invariant homogeneous convolution operator on G. If K is hypoelliptic, then (K) is injective on C for every nontrivial irreducible unitary representation of G (referred to as Rockland condition).展开更多
Sufficient conditions are presented for super/weak Poincare inequalities to hold for a class of hypoelliptic operators on noncompact manifolds. As applications, the essential spectrum and the convergence rate of the a...Sufficient conditions are presented for super/weak Poincare inequalities to hold for a class of hypoelliptic operators on noncompact manifolds. As applications, the essential spectrum and the convergence rate of the associated Markov semigroup are described for Gruschin type operators on R2 and Kohn-Laplacian type operators on the Heisenberg group.展开更多
In this paper we give a new proof regarding the regularity of solutions to hypoelliptic partial differential equations with constant coefficients. On the assumption of existence, we provide a spectral representation f...In this paper we give a new proof regarding the regularity of solutions to hypoelliptic partial differential equations with constant coefficients. On the assumption of existence, we provide a spectral representation for the solution and use this spectral representation to deduce regularity results. By exploiting analyticity properties of the terms within the spectral representation, we are able to give simple estimates for the size of the derivatives of the solutions and interpret them in terms of Gevrey classes.展开更多
We consider a Fokker-Planck operator with electric potential and electromagnetic fields.We establish the sharp weighted and subelliptic estimates,involving the control of the derivatives of electric potential and elec...We consider a Fokker-Planck operator with electric potential and electromagnetic fields.We establish the sharp weighted and subelliptic estimates,involving the control of the derivatives of electric potential and electromagnetic fields.Our proof relies on a localization argument as well as a careful calculation on commutators.展开更多
We obtained the Cα continuity for weak solutions of a class of ultraparabolic equations with measurable coeffcients of the form δt u = δx（a（x, y, t）δx u） ＋ b0（x, y, t）δxu ＋ b（x, y, t）δyu, which general...We obtained the Cα continuity for weak solutions of a class of ultraparabolic equations with measurable coeffcients of the form δt u = δx（a（x, y, t）δx u） ＋ b0（x, y, t）δxu ＋ b（x, y, t）δyu, which generalized our recent results on KFP equations.展开更多
Let Q be a bounded open domain in R^n with smooth boundaryаΩ.Let X=(X1,X2…,Xm)be a system of general Grushin type vector fields defined onΩand the boundaryаΩis non-characteristic for X.For△x=∑j=1^mXj^2,we den...Let Q be a bounded open domain in R^n with smooth boundaryаΩ.Let X=(X1,X2…,Xm)be a system of general Grushin type vector fields defined onΩand the boundaryаΩis non-characteristic for X.For△x=∑j=1^mXj^2,we denoteλk as the k-th eigenvalue for the bi-subelliptic operator△X2^2 onΩ.In this paper,by using the sharp sub-elliptic estimates and maximally hypoeliptic estimates,we give the optimal lower bound estimates ofλk for the operatork△X^2.展开更多
The one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent viscosity and drag force coefficients is investigated in the present paper.The existence,uniqueness,and regularity of g...The one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent viscosity and drag force coefficients is investigated in the present paper.The existence,uniqueness,and regularity of global weak solution to the initial value problem for general initial data are established in spatial periodic domain.Moreover,the long time behavior of the weak solution is analyzed.It is shown that as the time grows,the distribution function of the particles converges to the global Maxwellian,and both the fluid velocity and the macroscopic velocity of the particles converge to the same speed.展开更多
We study here a class of pseudodifferential operators with weighted symbols of Shubin type. First, we develop the basic elements of the pseudodifferential calculus for these operators, proving in particular a result o...We study here a class of pseudodifferential operators with weighted symbols of Shubin type. First, we develop the basic elements of the pseudodifferential calculus for these operators, proving in particular a result of L^P-boundedness. Then we derive regularity results in the frame of suitably defined functional spaces of Sobolev type.展开更多
文摘This paper deals with the Gevreg-hypoellipticity for a class of totally characteristic operators with the elliptic condition and the discrete boundary spectrum condition respectively.
文摘In this paper,we use the method of representation of Lie group to study a class of nonhomoge- neous convolution operator on the nilpotent Lie group H^M×R^k,and give a criteerion of their hypoellipticity.
基金Project supported by the National Natural Science Foundation of China.
文摘Let G be a connected and simply-connected step two homogeneous group, K be a left invariant homogeneous convolution operator on G. If K is hypoelliptic, then (K) is injective on C for every nontrivial irreducible unitary representation of G (referred to as Rockland condition).
基金Supported by the WIMCS,Creative Research Group Fund of the National Natural Science Foundation of China (No.10721091)the 973-Project
文摘Sufficient conditions are presented for super/weak Poincare inequalities to hold for a class of hypoelliptic operators on noncompact manifolds. As applications, the essential spectrum and the convergence rate of the associated Markov semigroup are described for Gruschin type operators on R2 and Kohn-Laplacian type operators on the Heisenberg group.
文摘In this paper we give a new proof regarding the regularity of solutions to hypoelliptic partial differential equations with constant coefficients. On the assumption of existence, we provide a spectral representation for the solution and use this spectral representation to deduce regularity results. By exploiting analyticity properties of the terms within the spectral representation, we are able to give simple estimates for the size of the derivatives of the solutions and interpret them in terms of Gevrey classes.
基金supported by NSFC(Grant Nos.11961160716,11871054,11771342)Fundamental Research Funds for the Central Universities(Grant No.2042020kf0210)。
文摘We consider a Fokker-Planck operator with electric potential and electromagnetic fields.We establish the sharp weighted and subelliptic estimates,involving the control of the derivatives of electric potential and electromagnetic fields.Our proof relies on a localization argument as well as a careful calculation on commutators.
基金partially supported by the NSF of China (10325104)
文摘We obtained the Cα continuity for weak solutions of a class of ultraparabolic equations with measurable coeffcients of the form δt u = δx（a（x, y, t）δx u） ＋ b0（x, y, t）δxu ＋ b（x, y, t）δyu, which generalized our recent results on KFP equations.
基金supported by National Natural Science Foundation of China (Grants Nos. 11631011 and 11626251)
文摘Let Q be a bounded open domain in R^n with smooth boundaryаΩ.Let X=(X1,X2…,Xm)be a system of general Grushin type vector fields defined onΩand the boundaryаΩis non-characteristic for X.For△x=∑j=1^mXj^2,we denoteλk as the k-th eigenvalue for the bi-subelliptic operator△X2^2 onΩ.In this paper,by using the sharp sub-elliptic estimates and maximally hypoeliptic estimates,we give the optimal lower bound estimates ofλk for the operatork△X^2.
基金supported by National Natural Science Foundation of China(Nos.11931010,11671384,11871047)by the key research project of Academy for Multidisciplinary Studies,Capital Normal Universityby the Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds(No.007/20530290068).
文摘The one-dimensional compressible Navier-Stokes-Vlasov-Fokker-Planck system with density-dependent viscosity and drag force coefficients is investigated in the present paper.The existence,uniqueness,and regularity of global weak solution to the initial value problem for general initial data are established in spatial periodic domain.Moreover,the long time behavior of the weak solution is analyzed.It is shown that as the time grows,the distribution function of the particles converges to the global Maxwellian,and both the fluid velocity and the macroscopic velocity of the particles converge to the same speed.
文摘We study here a class of pseudodifferential operators with weighted symbols of Shubin type. First, we develop the basic elements of the pseudodifferential calculus for these operators, proving in particular a result of L^P-boundedness. Then we derive regularity results in the frame of suitably defined functional spaces of Sobolev type.
