For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2)...For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2),Weak L^1(R^2).This extends the theorem of Stein & Wainger and the theo- rem of Weinberg.展开更多
We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas ...We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas for derivatives,for the solutions u=u(t,x)to parabolic equations of the form∂tu−aij(t)∂iju+u=f,t∈,x∈n and for the Cauchy problem{∂tv−aij(t)∂ijv+v=fv(0,x)forfort>0,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation∂tu−Δu+u=f.展开更多
This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. B...This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.展开更多
We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise.Owing to the damping term,under appropriate conditions on the nonlinearity,the solution admits a unique invariant distrib...We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise.Owing to the damping term,under appropriate conditions on the nonlinearity,the solution admits a unique invariant distribution.We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution,using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively.We prove that the considered numerical schemes also admit unique invariant distributions,and we prove error estimates between the approximate and exact invariant distributions,with identification of the orders of convergence.To the best of our knowledge this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.展开更多
In this paper, we first introduce interpolation operator of projection type in three dimen- sions, from which we derive weak estimates for tricubic block finite elements. Then using the estimate for the W 2, 1-seminor...In this paper, we first introduce interpolation operator of projection type in three dimen- sions, from which we derive weak estimates for tricubic block finite elements. Then using the estimate for the W 2, 1-seminorm of the discrete derivative Green's function and the weak estimates, we show that the tricubic block finite element solution uh and the tricubic interpolant of projection type Πh3u have superclose gradient in the pointwise sense of the L∞-norm. Finally, this supercloseness is applied to superconvergence analysis, and the global superconvergence of the finite element approximation is derived.展开更多
Let （X, μ） be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L^P,...Let （X, μ） be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L^P,q（X） with p, q ￡ （0, ∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.： Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ￡ [1, ∞） and q ￡ [l,∞）, was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].展开更多
文摘For the plane curves Γ,the maximal operator associated to it is defined by Mf(x)=sup|∫f(x-Γ(t))(r^(-1)t)r^(-1)dt| where is a Schwartz function.For a certain class of curves in R^2,M is shown to bounded on (H(R^2),Weak L^1(R^2).This extends the theorem of Stein & Wainger and the theo- rem of Weinberg.
基金supported by Simons Foundation(Grant No.580911(Stinga))Ministerio de Economía y Competitividad/Fondo Europeo de Desarrollo Regional from Government of Spain(Grant No.MTM2015-66157-C2-1-P(Torrea))。
文摘We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 1<p,q<∞and 0<α<1,weighted mixed weak-type estimates for q=1,L∞t(Lpx)−BMOt(W2,px)and L∞t(Cαx)−BMOt(C2,xx),and a.e.pointwise formulas for derivatives,for the solutions u=u(t,x)to parabolic equations of the form∂tu−aij(t)∂iju+u=f,t∈,x∈n and for the Cauchy problem{∂tv−aij(t)∂ijv+v=fv(0,x)forfort>0,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation∂tu−Δu+u=f.
基金the NSF-Guangdong China(04010473)Jinan University Foundation(51204033)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry(No.2005-383)
文摘This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.
基金supported by the projects ADA(Grant No.ANR-19-CE40-0019-02)and SIMALIN(Grant No.ANR-19-CE40-0016)operated by the French National Research Agencysupported by the NSF of China(Grant Nos.11971488,12371417)supported by the China Scholarship Council(Grant No.202206370085)。
文摘We study a class of stochastic semilinear damped wave equations driven by additive Wiener noise.Owing to the damping term,under appropriate conditions on the nonlinearity,the solution admits a unique invariant distribution.We apply semi-discrete and fully-discrete methods in order to approximate this invariant distribution,using a spectral Galerkin method and an exponential Euler integrator for spatial and temporal discretization respectively.We prove that the considered numerical schemes also admit unique invariant distributions,and we prove error estimates between the approximate and exact invariant distributions,with identification of the orders of convergence.To the best of our knowledge this is the first result in the literature concerning numerical approximation of invariant distributions for stochastic damped wave equations.
基金supported by Natural Science Foundation of Ningbo City (Grant No. 2008A610020)National Natural Science Foundation of China (Grant No. 10671065)the Scientific Research Fund of Hunan Provincial Education Department (Grant Nos. 07C576, 03C212)
文摘In this paper, we first introduce interpolation operator of projection type in three dimen- sions, from which we derive weak estimates for tricubic block finite elements. Then using the estimate for the W 2, 1-seminorm of the discrete derivative Green's function and the weak estimates, we show that the tricubic block finite element solution uh and the tricubic interpolant of projection type Πh3u have superclose gradient in the pointwise sense of the L∞-norm. Finally, this supercloseness is applied to superconvergence analysis, and the global superconvergence of the finite element approximation is derived.
基金The second author is supported by the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China （Grant No. 10XNF090） the third author is supported by National Natural Science Foundation of China （Grant No. 10871025）
文摘Let （X, μ） be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L^P,q（X） with p, q ￡ （0, ∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.： Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ￡ [1, ∞） and q ￡ [l,∞）, was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].
