It was proved by Bahouri et al.[9]that the Schrodinger equation on the Heisenberg group H^(d),involving the sublaplacian,is an example of a totally non-dispersive evolution equation:for this reason global dispersive e...It was proved by Bahouri et al.[9]that the Schrodinger equation on the Heisenberg group H^(d),involving the sublaplacian,is an example of a totally non-dispersive evolution equation:for this reason global dispersive estimates cannot hold.This paper aims at establishing local dispersive estimates on H^(d) for the linear Schrodinger equation,by a refined study of the Schrodinger ker-nel St on H^(d).The sharpness of these estimates is discussed through several examples.Our approach,based on the explicit formula of the heat kernel on H^(d) derived by Gaveau[19],is achieved by combining complex analysis and Fourier-Heisenberg tools.As a by-product of our results we establish local Stri-chartz estimates and prove that the kernel St concentrates on quantized hori-zontal hyperplanes of H^(d).展开更多
We prove an estimate for Donaldson's Q-operator on a prequantized compact symplectic manifold.This estimate is an ingredient in the recent result of Keller and Lejmi(2017) about a symplectic generalization of Dona...We prove an estimate for Donaldson's Q-operator on a prequantized compact symplectic manifold.This estimate is an ingredient in the recent result of Keller and Lejmi(2017) about a symplectic generalization of Donaldson's lower bound for the L^2-norm of the Hermitian scalar curvature.展开更多
We study the dynamics of a piecewise （in time） distributed optimal con- trol problem for Generalized MHD equations which model velocity tracking coupled to magnetic field over time. The long-time behavior of solutio...We study the dynamics of a piecewise （in time） distributed optimal con- trol problem for Generalized MHD equations which model velocity tracking coupled to magnetic field over time. The long-time behavior of solutions for an optimal dis- tributed control problem associated with the Generalized MHD equations is studied. First, a quasi-optimal solution for the Generalized MHD equations is constructed; this quasi-optimal solution possesses the decay （in time） properties. Then, some prelimi- nary estimates for the long-time behavior of all solutions of Generalized MHD equa- tions are derived. Next, the existence of a solution of optimal control problem is proved also optimality system is derived. Finally, the long-time decay properties for the opti- mal solutions is established.展开更多
文摘It was proved by Bahouri et al.[9]that the Schrodinger equation on the Heisenberg group H^(d),involving the sublaplacian,is an example of a totally non-dispersive evolution equation:for this reason global dispersive estimates cannot hold.This paper aims at establishing local dispersive estimates on H^(d) for the linear Schrodinger equation,by a refined study of the Schrodinger ker-nel St on H^(d).The sharpness of these estimates is discussed through several examples.Our approach,based on the explicit formula of the heat kernel on H^(d) derived by Gaveau[19],is achieved by combining complex analysis and Fourier-Heisenberg tools.As a by-product of our results we establish local Stri-chartz estimates and prove that the kernel St concentrates on quantized hori-zontal hyperplanes of H^(d).
基金supported by National Natural Science Foundation of China(Grant Nos.11401232 and 11528103)Agence nationale de la recherche(Grant No.ANR-14-CE25-0012-01)funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative and Deutsche Forschungsgemeinschaft Funded Project Sonderforschungsbereich Transregio 191
文摘We prove an estimate for Donaldson's Q-operator on a prequantized compact symplectic manifold.This estimate is an ingredient in the recent result of Keller and Lejmi(2017) about a symplectic generalization of Donaldson's lower bound for the L^2-norm of the Hermitian scalar curvature.
文摘We study the dynamics of a piecewise （in time） distributed optimal con- trol problem for Generalized MHD equations which model velocity tracking coupled to magnetic field over time. The long-time behavior of solutions for an optimal dis- tributed control problem associated with the Generalized MHD equations is studied. First, a quasi-optimal solution for the Generalized MHD equations is constructed; this quasi-optimal solution possesses the decay （in time） properties. Then, some prelimi- nary estimates for the long-time behavior of all solutions of Generalized MHD equa- tions are derived. Next, the existence of a solution of optimal control problem is proved also optimality system is derived. Finally, the long-time decay properties for the opti- mal solutions is established.
