In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilin...In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.展开更多
We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type.Ourfirst existence and uniqueness result concerns those such manifolds which are asymptotically local...We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type.Ourfirst existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic.In this context,our result requires only a partial C2 decay of the metric,namely the full decay of the metric in C1 and the decay of the scalar curvature.In particular,no decay of the Ricci curvature is assumed.In our second result we establish that a local volume ratio condition,when combined with negativity of the scalar curvature at infinity,is sufficient for existence of a solution.Our volume ratio condition appears tight.This paper is based on the DPhil thesis of thefirst author.展开更多
We prove the positive energy conjecture for a class of asymptotically Horowitz-Myers(AHM)metrics on R^(2)×T^(n-2).This generalizes the previous results of Barzegar et al.(2020)as well as Liang and Zhang(2020).
文摘In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.
基金supported by the EPSRC Centre for Doctoral Training in Partial Differential Equations(grant number EP/L015811/1).
文摘We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type.Ourfirst existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic.In this context,our result requires only a partial C2 decay of the metric,namely the full decay of the metric in C1 and the decay of the scalar curvature.In particular,no decay of the Ricci curvature is assumed.In our second result we establish that a local volume ratio condition,when combined with negativity of the scalar curvature at infinity,is sufficient for existence of a solution.Our volume ratio condition appears tight.This paper is based on the DPhil thesis of thefirst author.
基金supported by National Natural Science Foundation of China (Grant No. 11731001)the special foundation for Junwu and Guangxi Ba Gui Scholarsthe Guangdong Basic and Applied Basic Research Foundation (Grant No. 2021A1515010379)
文摘We prove the positive energy conjecture for a class of asymptotically Horowitz-Myers(AHM)metrics on R^(2)×T^(n-2).This generalizes the previous results of Barzegar et al.(2020)as well as Liang and Zhang(2020).
