We show the existence of Holder continuous periodic weak solutions of the 2D Boussinesq equation with thermal diffusion which satisfy the prescribed kinetic energy.More precisely,for any smooth e(t):[0,1]→R+andε∈(0...We show the existence of Holder continuous periodic weak solutions of the 2D Boussinesq equation with thermal diffusion which satisfy the prescribed kinetic energy.More precisely,for any smooth e(t):[0,1]→R+andε∈(0,110),there exist v∈C 110−ε([0,1]×T2)andθ∈C 1,120−εt 2 C 2,1 x 10−ε([0,1]×T2),which satisfy(1.1)in the sense of distribution and e(t)=ˆT2|v(t,x)|2 dx,∀t∈[0,1].展开更多
In this paper,I consider the Hölder continuity of the Lyapunov exponent for a quasi-periodic Szegö cocycle with weak Liouville frequency.I extend the existing results about the regularity of the Lyapunov exp...In this paper,I consider the Hölder continuity of the Lyapunov exponent for a quasi-periodic Szegö cocycle with weak Liouville frequency.I extend the existing results about the regularity of the Lyapunov exponent from the Schrödinger cocycle in[24]to a Szegö cocycle.展开更多
We show the existence of dissipative Hhlder continuous solutions of the Boussi- nesq equations. More precise, for anyβ∈ （0, 1/5）, a time interval [0, T] and any given smooth energy profile e ： [0, T] → （0, ∞...We show the existence of dissipative Hhlder continuous solutions of the Boussi- nesq equations. More precise, for anyβ∈ （0, 1/5）, a time interval [0, T] and any given smooth energy profile e ： [0, T] → （0, ∞）, there exist a weak solution （v, θ） of the 3d Boussinesq equations such that （v, 8） ∈ Cβ（T3 × [0, T]） with e（t） = ∫T3 ｜v（x, t）｜2dx for all t ∈ [0, T]. This extend the result of [2] about Onsager＇s conjecture into Boussinesq equation and improve our previous result in [30].展开更多
We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with tem...We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density μ(ξ). We assume that ■ and ■ , where αi, i = 1, · · ·, d(or α) can take negative value.展开更多
In this paper, we study the regularity of mild solution for the following fractional abstract Cauchy problem Dt αu(t)=Au(t)+f(t), t ∈ (0,T] u(0)= x0 on a Banach space X with order α ∈ (0,1), where the fractional d...In this paper, we study the regularity of mild solution for the following fractional abstract Cauchy problem Dt αu(t)=Au(t)+f(t), t ∈ (0,T] u(0)= x0 on a Banach space X with order α ∈ (0,1), where the fractional derivative is understood in the sense of Caputo fractional derivatives. We show that if A generates an analytic α-times resolvent family on X and f ∈ Lp ([0,T];X) for some p > 1/α, then the mild solution to the above equation is in Cα-1/p[ò,T] for every ò > 0. Moreover, if f is H?lder continuous, then so are the Dt αu(t) and Au(t).展开更多
In this short paper,we remove the restrictionγ∈(1,3]that was used in the paper"The rate of convergence of the viscosity method for a nonlinear hyperbolic system"(Nonlinear Analysis,1999,38:435-445)and obta...In this short paper,we remove the restrictionγ∈(1,3]that was used in the paper"The rate of convergence of the viscosity method for a nonlinear hyperbolic system"(Nonlinear Analysis,1999,38:435-445)and obtain a global Holder continuous solution and the convergent rate of the viscosity method for the Cauchy problem of the variant nonisentropic system of polytropic gas for any adiabatic exponentγ>1.展开更多
Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quan...Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.展开更多
Let A and B satisfy the structural conditions (2), the local Holder continuity interior to Q = G X (0, T) is proved for the generalized solutions of quasilinear parabolic equations as follows: u(t) - divA(x, t, u, del...Let A and B satisfy the structural conditions (2), the local Holder continuity interior to Q = G X (0, T) is proved for the generalized solutions of quasilinear parabolic equations as follows: u(t) - divA(x, t, u, del u) + B(x, t, U, del u) = 0.展开更多
On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Holder continuous property of some singular integral operat...On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Holder continuous property of some singular integral operators with one integral variable. Then we divide a singular integral operator with two variables into three parts and prove its Holder continuous property on the boundary.展开更多
Convergence of modified truncated Euler-Maruyama(MTEM)method for stochastic differential equations(SDEs)with(1/2+α)-Holder continuous diffusion coefficients are investigated in this paper.We prove that the MTEM metho...Convergence of modified truncated Euler-Maruyama(MTEM)method for stochastic differential equations(SDEs)with(1/2+α)-Holder continuous diffusion coefficients are investigated in this paper.We prove that the MTEM method for SDE converges to the exact solution in L9 sense under given conditions.Two examples are provided to support our conclusions.展开更多
In this paper,we furst construct a claas of fraetal funerions by means of b-adic fraction andinfinite series expressions.Then we investigate the fractal dimensions of the graphs of these funcrionsand Holder continuity...In this paper,we furst construct a claas of fraetal funerions by means of b-adic fraction andinfinite series expressions.Then we investigate the fractal dimensions of the graphs of these funcrionsand Holder continuity.Some of results of dimensions are obtained.展开更多
This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covarianc...This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.展开更多
In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equ...In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.展开更多
The sufficient conditions of Holder continuity of two kinds of fractal interpolation functions defined by IFS (Iterated Function System) were obtained. The sufficient and necessary condition for its differentiability ...The sufficient conditions of Holder continuity of two kinds of fractal interpolation functions defined by IFS (Iterated Function System) were obtained. The sufficient and necessary condition for its differentiability was proved. Its derivative was a fractal interpolation function generated by the associated IFS, if it is differentiable.展开更多
Well-Posedness for McKean-Vlasov SDEs Driven by Multiplicative Stable Noises Changsong Deng Xi ng Huang Abstract We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric Q-stable Levy proc...Well-Posedness for McKean-Vlasov SDEs Driven by Multiplicative Stable Noises Changsong Deng Xi ng Huang Abstract We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric Q-stable Levy processes(1/2<α≤1),where the drift coefficient is Holder continuous in space variable,while the noise coeficient is Lipscitz continuous in space variable,and both of them satisfy the Lipschitz condition in distribution variable with respect to Wasserstein distance.If the drift coefficient does not depend on distribution variable,our methodology developed in this paper applies to the caseαe(0,1].The main tool relies on heat kernel estimates for(distribution independent)stable SDEs and Banach's fixed point theorem.展开更多
Let （Zt）t≥o be a one-dimensional symmetric α-stable process with α ∈ （0, 2）, and let a be a bounded （from above and from below） and 1/（α V 1）- Holder continuous function on R. Consider the stochastic diff...Let （Zt）t≥o be a one-dimensional symmetric α-stable process with α ∈ （0, 2）, and let a be a bounded （from above and from below） and 1/（α V 1）- Holder continuous function on R. Consider the stochastic differential equation dXt = σ（Xt-）dZt, which admits a unique strong solution. By using the splitting technique and the coupling method, we derive the HSlder continuity of the associated semigroup.展开更多
In this paper,we establish quantitative Green’s function estimates for some higher-dimensional lattice quasi-periodic(QP)Schrodinger operators.The resonances in the estimates can be described via a pair of symmetric ...In this paper,we establish quantitative Green’s function estimates for some higher-dimensional lattice quasi-periodic(QP)Schrodinger operators.The resonances in the estimates can be described via a pair of symmetric zeros of certain functions,and the estimates apply to the sub-exponential-type non-resonance conditions.As the application of quantitative Green’s function estimates,we prove both the arithmetic version of Anderson localization and the finite volume version of(1/2-)-Holder continuity of the integrated density of states(IDS)for such QP Schrodinger operators.This gives an affirmative answer to Bourgain’s problem in Bourgain(2000).展开更多
The authors are concerned with a class of one-dimensional stochastic Anderson models with double-parameter fractional noises, whose differential operators are fractional. A unique solution for the model in some approp...The authors are concerned with a class of one-dimensional stochastic Anderson models with double-parameter fractional noises, whose differential operators are fractional. A unique solution for the model in some appropriate Hilbert space is constructed. Moreover, the Lyapunov exponent of the solution is estimated, and its HSlder continuity is studied. On the other hand, the absolute continuity of the solution is also discussed.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11971464)supported by National Natural Science Foundation of China(Grant No.11901349)supported by National Natural Science Foundation of China(Grant Nos.11471320 and 11631008)。
文摘We show the existence of Holder continuous periodic weak solutions of the 2D Boussinesq equation with thermal diffusion which satisfy the prescribed kinetic energy.More precisely,for any smooth e(t):[0,1]→R+andε∈(0,110),there exist v∈C 110−ε([0,1]×T2)andθ∈C 1,120−εt 2 C 2,1 x 10−ε([0,1]×T2),which satisfy(1.1)in the sense of distribution and e(t)=ˆT2|v(t,x)|2 dx,∀t∈[0,1].
文摘In this paper,I consider the Hölder continuity of the Lyapunov exponent for a quasi-periodic Szegö cocycle with weak Liouville frequency.I extend the existing results about the regularity of the Lyapunov exponent from the Schrödinger cocycle in[24]to a Szegö cocycle.
基金partially supported by the NSFC(11471320 and 11631008)
文摘We show the existence of dissipative Hhlder continuous solutions of the Boussi- nesq equations. More precise, for anyβ∈ （0, 1/5）, a time interval [0, T] and any given smooth energy profile e ： [0, T] → （0, ∞）, there exist a weak solution （v, θ） of the 3d Boussinesq equations such that （v, 8） ∈ Cβ（T3 × [0, T]） with e（t） = ∫T3 ｜v（x, t）｜2dx for all t ∈ [0, T]. This extend the result of [2] about Onsager＇s conjecture into Boussinesq equation and improve our previous result in [30].
基金supported by an NSERC granta startup fund of University of Albertasupported by Martin Hairer’s Leverhulme Trust leadership award
文摘We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density μ(ξ). We assume that ■ and ■ , where αi, i = 1, · · ·, d(or α) can take negative value.
文摘In this paper, we study the regularity of mild solution for the following fractional abstract Cauchy problem Dt αu(t)=Au(t)+f(t), t ∈ (0,T] u(0)= x0 on a Banach space X with order α ∈ (0,1), where the fractional derivative is understood in the sense of Caputo fractional derivatives. We show that if A generates an analytic α-times resolvent family on X and f ∈ Lp ([0,T];X) for some p > 1/α, then the mild solution to the above equation is in Cα-1/p[ò,T] for every ò > 0. Moreover, if f is H?lder continuous, then so are the Dt αu(t) and Au(t).
基金supported by the National Natural Science Foundation of China(12071409)。
文摘In this short paper,we remove the restrictionγ∈(1,3]that was used in the paper"The rate of convergence of the viscosity method for a nonlinear hyperbolic system"(Nonlinear Analysis,1999,38:435-445)and obtain a global Holder continuous solution and the convergent rate of the viscosity method for the Cauchy problem of the variant nonisentropic system of polytropic gas for any adiabatic exponentγ>1.
基金supported by the NSF of Hebei Province(A2022208007)the NSF of China(11571089,11871191)the NSF of Henan Province(222300420397)。
文摘Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.
文摘Let A and B satisfy the structural conditions (2), the local Holder continuity interior to Q = G X (0, T) is proved for the generalized solutions of quasilinear parabolic equations as follows: u(t) - divA(x, t, u, del u) + B(x, t, U, del u) = 0.
基金Supported by the National Natural Science Foundation of China (10771049, 10801043)the Hebei Natural Science Foundation (A2007000225, A2010000346)
文摘On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Holder continuous property of some singular integral operators with one integral variable. Then we divide a singular integral operator with two variables into three parts and prove its Holder continuous property on the boundary.
基金supported by the Natural Science Foundation of Beijing Municipality(Grant No.1192013).
文摘Convergence of modified truncated Euler-Maruyama(MTEM)method for stochastic differential equations(SDEs)with(1/2+α)-Holder continuous diffusion coefficients are investigated in this paper.We prove that the MTEM method for SDE converges to the exact solution in L9 sense under given conditions.Two examples are provided to support our conclusions.
基金Supported by the Scientific Research Foundation of Huaibei Coal Industry Teaccher's Colhge(00061).
文摘In this paper,we furst construct a claas of fraetal funerions by means of b-adic fraction andinfinite series expressions.Then we investigate the fractal dimensions of the graphs of these funcrionsand Holder continuity.Some of results of dimensions are obtained.
基金supported by an NSERC granta startup fund of University of Alberta
文摘This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.
基金supported by an NSERC granta startup fund of University of Albertasupported by the NSF grant DMS1613163
文摘In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two differen t met hods, respectively, based on variance compu tations and on pat h-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.
文摘The sufficient conditions of Holder continuity of two kinds of fractal interpolation functions defined by IFS (Iterated Function System) were obtained. The sufficient and necessary condition for its differentiability was proved. Its derivative was a fractal interpolation function generated by the associated IFS, if it is differentiable.
文摘Well-Posedness for McKean-Vlasov SDEs Driven by Multiplicative Stable Noises Changsong Deng Xi ng Huang Abstract We establish the well-posedness for a class of McKean-Vlasov SDEs driven by symmetric Q-stable Levy processes(1/2<α≤1),where the drift coefficient is Holder continuous in space variable,while the noise coeficient is Lipscitz continuous in space variable,and both of them satisfy the Lipschitz condition in distribution variable with respect to Wasserstein distance.If the drift coefficient does not depend on distribution variable,our methodology developed in this paper applies to the caseαe(0,1].The main tool relies on heat kernel estimates for(distribution independent)stable SDEs and Banach's fixed point theorem.
基金The authors were indebted to the referees for their helpful comments and careful corrections. The first author＇s work was supported by the Key Laboratory of Random Complex Structures and Data Sciences, Chinese Academy of Sciences （2008DP173182）, the National Natural Science Foundation of China （Grant No. 11571347）, and Academy of Mathematics and Systems Science （Y129161ZZ1）. The second author＇s work was supported by the National Natural Science Foundation of China （Grant Nos. 11201073 and 11522106）, the National Science Foundation of Fujian Province （2015J01003）, and the Program for Nonlinear Analysis and Its Applications （IRTL1206）.
文摘Let （Zt）t≥o be a one-dimensional symmetric α-stable process with α ∈ （0, 2）, and let a be a bounded （from above and from below） and 1/（α V 1）- Holder continuous function on R. Consider the stochastic differential equation dXt = σ（Xt-）dZt, which admits a unique strong solution. By using the splitting technique and the coupling method, we derive the HSlder continuity of the associated semigroup.
基金supported by National Natural Science Foundation of China(Grant No.12271380)supported by National Natural Science Foundation of China(Grant Nos.12171010 and 12288101)National Key R&D Program(Grant No.2021YFA1001600)。
文摘In this paper,we establish quantitative Green’s function estimates for some higher-dimensional lattice quasi-periodic(QP)Schrodinger operators.The resonances in the estimates can be described via a pair of symmetric zeros of certain functions,and the estimates apply to the sub-exponential-type non-resonance conditions.As the application of quantitative Green’s function estimates,we prove both the arithmetic version of Anderson localization and the finite volume version of(1/2-)-Holder continuity of the integrated density of states(IDS)for such QP Schrodinger operators.This gives an affirmative answer to Bourgain’s problem in Bourgain(2000).
基金supported by the National Natural Science Foundation of China (No. 10871103)
文摘The authors are concerned with a class of one-dimensional stochastic Anderson models with double-parameter fractional noises, whose differential operators are fractional. A unique solution for the model in some appropriate Hilbert space is constructed. Moreover, the Lyapunov exponent of the solution is estimated, and its HSlder continuity is studied. On the other hand, the absolute continuity of the solution is also discussed.
