In this work,we propose a low-regularity Fourier integrator with almost mass conservation to solve the Davey-StewartsonⅡsystem(hyperbolic-elliptic case).Arbitrary order mass convergence could be achieved by the suita...In this work,we propose a low-regularity Fourier integrator with almost mass conservation to solve the Davey-StewartsonⅡsystem(hyperbolic-elliptic case).Arbitrary order mass convergence could be achieved by the suitable addition of correction terms,while keeping the first order accuracy in H~γ×H^(γ+1)for initial data in H^(γ+1)×H^(γ+1)withγ>1.The main theorem is that,up to some fixed time T,there exist constantsτ_(0)and C depending only on T and‖u‖_(L^(∞)((0,T);H^(γ+1)))such that,for any 0<τ≤τ_(0),we have that‖u(t_(n),·)-u^(n)‖H_γ≤C_(τ),‖v(t_(n),·)-v^(n)‖_(Hγ+1)≤C_(τ),where u^(n)and v^(n)denote the numerical solutions at t_(n)=nτ.Moreover,the mass of the numerical solution M(u^(n))satisfies that|M(u^(n))-M(u_0)|≤Cτ~5.展开更多
In this paper,we study Ricci flow on compact manifolds with a continuous initial metric.It was known from Simon(2002)that the Ricci flow exists for a short time.We prove that the scalar curvature lower bound is preser...In this paper,we study Ricci flow on compact manifolds with a continuous initial metric.It was known from Simon(2002)that the Ricci flow exists for a short time.We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in the distributional sense provided that the initial metric is W^(1,p) for some n<p∞.As an application,we use this result to study the relation between the Yamabe invariant and Ricci flat metrics.We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in the distributional sense,then the manifold is isometric to a Ricci flat manifold.展开更多
基金supported by the NSFC(11901120)supported by the NSFC(12171356)the Science and Technology Program of Guangzhou,China(2024A04J4027)。
文摘In this work,we propose a low-regularity Fourier integrator with almost mass conservation to solve the Davey-StewartsonⅡsystem(hyperbolic-elliptic case).Arbitrary order mass convergence could be achieved by the suitable addition of correction terms,while keeping the first order accuracy in H~γ×H^(γ+1)for initial data in H^(γ+1)×H^(γ+1)withγ>1.The main theorem is that,up to some fixed time T,there exist constantsτ_(0)and C depending only on T and‖u‖_(L^(∞)((0,T);H^(γ+1)))such that,for any 0<τ≤τ_(0),we have that‖u(t_(n),·)-u^(n)‖H_γ≤C_(τ),‖v(t_(n),·)-v^(n)‖_(Hγ+1)≤C_(τ),where u^(n)and v^(n)denote the numerical solutions at t_(n)=nτ.Moreover,the mass of the numerical solution M(u^(n))satisfies that|M(u^(n))-M(u_0)|≤Cτ~5.
基金supported by National Natural Science Foundation of China (Grant Nos.12125105 and 12071425)the Fundamental Research Funds for the Central Universities+1 种基金supported by National Natural Science Foundation of China (Grant Nos.11971424 and 12031017)supported by National Natural Science Foundation of China (Grant No.11971424)。
文摘In this paper,we study Ricci flow on compact manifolds with a continuous initial metric.It was known from Simon(2002)that the Ricci flow exists for a short time.We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in the distributional sense provided that the initial metric is W^(1,p) for some n<p∞.As an application,we use this result to study the relation between the Yamabe invariant and Ricci flat metrics.We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in the distributional sense,then the manifold is isometric to a Ricci flat manifold.
