Recently Guo introduced integrated Meyer -Konig and Zeller operators and studied the rate of convergence for function of bounded variation. In this note we give a sharp estimate for these operators.
Inspired by Durfee Conjecture in singularity theory, Yau formulated the Yau number theoretic conjecture(see Conjecture 1.3) which gives a sharp polynomial upper bound of the number of positive integral points in an n-...Inspired by Durfee Conjecture in singularity theory, Yau formulated the Yau number theoretic conjecture(see Conjecture 1.3) which gives a sharp polynomial upper bound of the number of positive integral points in an n-dimensional(n≥3) polyhedron. It is well known that getting the estimate of integral points in the polyhedron is equivalent to getting the estimate of the de Bruijn function ψ(x, y), which is important and has a number of applications to analytic number theory and cryptography. We prove the Yau number theoretic conjecture for n = 6. As an application, we give a sharper estimate of function ψ(x, y) for 5≤y < 17, compared with the result obtained by Ennola.展开更多
In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-or...In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-order methods or the very small time stepsτ1=O(τ2)for the first level solution u1.This is,the first-order consistence of the first level solution u1 like BDF1(i.e.Euler scheme)as a starting point does not cause the loss of global temporal accuracy,and the ratios are updated to rk≤4.8645.展开更多
An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity p...An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations.The present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.展开更多
We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions ...We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions of the Cauchy problem for an n-dimensional incompressible Navier-Stokes equations.We also use the global smooth solution of the corresponding heat equation to approximate the global weak solutions of the incompressible Navier-Stokes equations.展开更多
Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u_0(x) and with the incompressible conditions ? · ...Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u_0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u_0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u_0∈ L1(Rn) ∩ L^2(Rn) and the external force f ∈ L^1(Rn× R+) ∩ L^1(R+, L^2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t > 0, where the dimension n ≥ 2, C > 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.展开更多
In this paper, we establish a sharp function estimate for the multilinear integral operators associated to the pseudo-differential operators. As the application, we obtain the L<sup>p</sup> (1 p norm ...In this paper, we establish a sharp function estimate for the multilinear integral operators associated to the pseudo-differential operators. As the application, we obtain the L<sup>p</sup> (1 p norm inequalities for the multilinear operators.展开更多
In the article we study the solution u(x,t)of the Cauchy problem of linear damped fractional wave equation.We prove that u(x,t)has some sharp boundedness estimates on the Triebel–Lizorkin space.The proof of the neces...In the article we study the solution u(x,t)of the Cauchy problem of linear damped fractional wave equation.We prove that u(x,t)has some sharp boundedness estimates on the Triebel–Lizorkin space.The proof of the necessity part is based on obtaining the precise asymptotic forms of the kernels of operators e^(-t)cosh(t√L)and e^(-t)sinh(t√L)/√L with L=1-|Δ|α,whereΔis the Laplacian,as well as the method of stationary phase.Additionally,we study the Riesz mean of the solution and show its convergence in the Triebel–Lizorkin space norm.展开更多
In this paper,we give four kinds of sharp estimates of two variants of bilinear Hausdorff operators on stratified groups,involving weighted Lebesgue spaces,classical Morrey spaces and central Morrey spaces.Meanwhile,s...In this paper,we give four kinds of sharp estimates of two variants of bilinear Hausdorff operators on stratified groups,involving weighted Lebesgue spaces,classical Morrey spaces and central Morrey spaces.Meanwhile,some necessary and sufficient conditions of boundness are obtained.展开更多
Let X and Y be two Banach spaces,and f:X→Y be a standard ε-isometry for some ε >= 0.In this paper,by using a recent theorem established by Cheng et al.(2013–2015),we show a sufficient condition guaranteeing the...Let X and Y be two Banach spaces,and f:X→Y be a standard ε-isometry for some ε >= 0.In this paper,by using a recent theorem established by Cheng et al.(2013–2015),we show a sufficient condition guaranteeing the following sharp stability inequality of f:There is a surjective linear operator T:Y→X of norm one so that ||T f(x)-x||<= 2ε,for all x∈X.As its application,we prove the following statements are equivalent for a standard ε-isometry f:X→Y:(i)lim inf_(t→∞) dist(ty,f(X))/|t|<1/2,for all y∈S_Y;(ii)τ(f)≡sup_(y∈S_Y) lim inf_(t→∞) dist(ty,f(X))/|t|=0;(iii)there is a surjective linear isometry U:X→Y so that || f(x)-Ux||<= 2ε,for all x∈X.This gives an affirmative answer to a question proposed by Vestfrid(2004,2015).展开更多
This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group Hn. The sharp bounds for the strong type （p,p） （1 〈 p 〈 ∞） estimates of n- dimensional Hausdorff operato...This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group Hn. The sharp bounds for the strong type （p,p） （1 〈 p 〈 ∞） estimates of n- dimensional Hausdorff operators on Hn are obtained. The sharp bounds for strong （p,p） estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on Hn. The weak type （p,p） （1 〈 p 〈 ∞） estimates are also obtained.展开更多
Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak soluti...Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations.· Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?· Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?· Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?· If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?· Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order m of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.展开更多
We provide the H2-regularity result of the solution ip and its first-order time derivative ipt and the second-order time derivative iptt for the complex Ginzburg-Landau equation with the Dirichlet or Neumann boundary ...We provide the H2-regularity result of the solution ip and its first-order time derivative ipt and the second-order time derivative iptt for the complex Ginzburg-Landau equation with the Dirichlet or Neumann boundary conditions.The analysis shows that these regularity results are uniform when t tends to ∞ and 0 and are dependent of the powers of ε^-1.展开更多
基金Research supported by Council of Scientific and Industrial Research, India under award no.9/143(163)/91-EER-
文摘Recently Guo introduced integrated Meyer -Konig and Zeller operators and studied the rate of convergence for function of bounded variation. In this note we give a sharp estimate for these operators.
基金the Start-Up Fund from Tsinghua University, Tsinghua University Initiative Scientific Research Program and National Natural Science Foundation of China (Grant No. 11401335)
文摘Inspired by Durfee Conjecture in singularity theory, Yau formulated the Yau number theoretic conjecture(see Conjecture 1.3) which gives a sharp polynomial upper bound of the number of positive integral points in an n-dimensional(n≥3) polyhedron. It is well known that getting the estimate of integral points in the polyhedron is equivalent to getting the estimate of the de Bruijn function ψ(x, y), which is important and has a number of applications to analytic number theory and cryptography. We prove the Yau number theoretic conjecture for n = 6. As an application, we give a sharper estimate of function ψ(x, y) for 5≤y < 17, compared with the result obtained by Ennola.
基金Natural Science Foundation of Hubei Province(2019CFA007)Supported by NSFC(11771035).
文摘In this note we announce the sharp error estimate of BDF2 scheme for linear diffusion reaction problem with variable time steps.Our analysis shows that the optimal second-order convergence does not require the high-order methods or the very small time stepsτ1=O(τ2)for the first level solution u1.This is,the first-order consistence of the first level solution u1 like BDF1(i.e.Euler scheme)as a starting point does not cause the loss of global temporal accuracy,and the ratios are updated to rk≤4.8645.
基金supported by the National Natural Science Foundation of China under grants 11771162,11771035,12171376 and 2020-JCJQ-ZD-029.
文摘An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial time.The weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations.The present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.
文摘We couple together existing ideas,existing results,special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions of the Cauchy problem for an n-dimensional incompressible Navier-Stokes equations.We also use the global smooth solution of the corresponding heat equation to approximate the global weak solutions of the incompressible Navier-Stokes equations.
文摘Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u_0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u_0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u_0∈ L1(Rn) ∩ L^2(Rn) and the external force f ∈ L^1(Rn× R+) ∩ L^1(R+, L^2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t > 0, where the dimension n ≥ 2, C > 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.
文摘In this paper, we establish a sharp function estimate for the multilinear integral operators associated to the pseudo-differential operators. As the application, we obtain the L<sup>p</sup> (1 p norm inequalities for the multilinear operators.
基金Supported by the National Key Research and Development Program of China(Grant No.2022YFA1005700)the Natural Science Foundation of Guangdong Province(Grant No.2023A1515012034)the National Natural Science Foundation of China(Grant Nos.12371105,11971295)。
文摘In the article we study the solution u(x,t)of the Cauchy problem of linear damped fractional wave equation.We prove that u(x,t)has some sharp boundedness estimates on the Triebel–Lizorkin space.The proof of the necessity part is based on obtaining the precise asymptotic forms of the kernels of operators e^(-t)cosh(t√L)and e^(-t)sinh(t√L)/√L with L=1-|Δ|α,whereΔis the Laplacian,as well as the method of stationary phase.Additionally,we study the Riesz mean of the solution and show its convergence in the Triebel–Lizorkin space norm.
基金supported by National Natural Science Foundation of China(Grant Nos.11471040 and 11761131002).
文摘In this paper,we give four kinds of sharp estimates of two variants of bilinear Hausdorff operators on stratified groups,involving weighted Lebesgue spaces,classical Morrey spaces and central Morrey spaces.Meanwhile,some necessary and sufficient conditions of boundness are obtained.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371296, 11401370 and 11471270)Ph D Programs Foundation of Ministry of Education of the Peoples Republic of China (Grant No. 20130121110032)+1 种基金Natural Science Foundation of Fujian Province (Grant No. 2015J01022)Fundamental Research Funds for the Central Universities (Grant No. 20720160010)
文摘Let X and Y be two Banach spaces,and f:X→Y be a standard ε-isometry for some ε >= 0.In this paper,by using a recent theorem established by Cheng et al.(2013–2015),we show a sufficient condition guaranteeing the following sharp stability inequality of f:There is a surjective linear operator T:Y→X of norm one so that ||T f(x)-x||<= 2ε,for all x∈X.As its application,we prove the following statements are equivalent for a standard ε-isometry f:X→Y:(i)lim inf_(t→∞) dist(ty,f(X))/|t|<1/2,for all y∈S_Y;(ii)τ(f)≡sup_(y∈S_Y) lim inf_(t→∞) dist(ty,f(X))/|t|=0;(iii)there is a surjective linear isometry U:X→Y so that || f(x)-Ux||<= 2ε,for all x∈X.This gives an affirmative answer to a question proposed by Vestfrid(2004,2015).
基金Supported by National Natural Science Foundation of China(Grant No.11201287)China Scholarship Council(Grant No.201406895019)
文摘This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group Hn. The sharp bounds for the strong type （p,p） （1 〈 p 〈 ∞） estimates of n- dimensional Hausdorff operators on Hn are obtained. The sharp bounds for strong （p,p） estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on Hn. The weak type （p,p） （1 〈 p 〈 ∞） estimates are also obtained.
文摘Consider the n-dimensional incompressible Navier-Stokes equations δ/(δt)u-α△u +(u ·△↓)u + △↓p = f(x, t), △↓· u = 0,△↓· f = 0,u(x, 0) = u0(x), △↓·u0=0.There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations.· Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?· Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?· Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?· If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?· Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order m of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.
基金supported by the Major Research and Development Program of China(Grant No.2016YFB0200901)the National Natural Science Foundation of China(Grant No.11771348).
文摘We provide the H2-regularity result of the solution ip and its first-order time derivative ipt and the second-order time derivative iptt for the complex Ginzburg-Landau equation with the Dirichlet or Neumann boundary conditions.The analysis shows that these regularity results are uniform when t tends to ∞ and 0 and are dependent of the powers of ε^-1.
