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DISCONTINUOUS SOLUTIONS IN L~∞ FOR HAMILTON-JACOBI EQUATIONS

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Chinese Annals of mathematics,Series B 2000年第2期21卷 165-186页

作者： CHENGUIQIANG SUBo ：departmentofmathematics NorthwesternUniversity2033SheridanRoadEvanstonIL60208-2730USA departmentofmathematics UniversityofWisconsinatMadisonMadisonWI53706-1380USA.

An approach is introduced to construct global discontinuous solutions in L~∞ for HamiltonJacobi equations. This approach allows the initial data only in L~∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discoatinuous solutions in L~∞. The existence of global discontinuous solutions in L~∞ is established. These solutions in L~∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed to examine the L~∞ stability of our L~∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated.

关键词： Hamilton-Jacobi equations Discontinuous solutions Profit functions Viscosity solutions Madman solutions Stability

A note on certain square functions on product spaces

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Science China mathematics 2006年第1期49卷 98-108页

作者： WANG Meng, CHEN Jiecheng2 & FAN Dashan department of mathematics, Zhejiang university (at Yuquan campus), Hangzhou 310027, China department of mathematics, Zhejiang university (at Xixi campus), Hangzhou 310028, China department of mathematics, university of wisconsin-Milwaukee, Milwaukee, wi53201, usa department of mathematics Zhejiang University [at Yuquan campus] Hangzhou 310027 China department of mathematics Zhejiang University [at Xixi campus] Hangzhou 310028 China department of mathematics University of Wisconsin-Milwaukee Milwaukee WI53201 USA

We study certain square functions on product spaces Rn × Rm, whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log+ L) away from Rn × wi and wi × Rm by means of polynomial distortions in the radial variable. As a model case, we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn × Rm)(P > 1) for Ω∈ e Llog+ L(Sn-1 × Sm-1) satisfying the cancellation condition.

关键词： square function, Marcinkiewicz integral, product space.

Gerbes and twisted orbifold quantum cohomology

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Science China mathematics 2008年第6期51卷 995-1016页

作者： PAN JianZhong~1 RUAN YongBin~(2,3) YIN XiaoQin~(1,3+) 1 Institute of mathematics,Academy of mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100080,China 2 department of mathematics,university of wisconsin-madison,wi 53.06,usa 3 department of mathematics,The Hong Kong university of Science and Technology,Hong Kong,China Institute of mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 China department of mathematics University of Wisconsin-Madison WI 53706 Madison USA department of mathematics The Hong Kong University of Science and Technology Hong Kong China

In this paper,we construct an orbifold quantum cohomology twisted by a flat gerbe. Then we compute these invariants in the case of a smooth manifold and a discrete torsion on a global quotient orbifold.

关键词： gerbe orbifold quantum cohomology 53D45

Littlewood-Paley operators with variable kernels

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Science China mathematics 2006年第5期49卷 639-650页

作者： CHEN Jiecheng, DING Yong & FAN Dashan department of mathematics, Zhejiang university (Xixi Campus), Hangzhou 310028, China School of Mathematical Sciences, Beijing Normal university, Beijing 100875, China department of mathematics, university of wisconsin-Milwaukee, Milwaukee wi 53201, usa department of mathematics, Central China Normal university, Wuhan 430074, China department of mathematics Zhejiang University [Xixi Campus] Hangzhou 310028 China School of Mathematical Sciences Beijing Normal University Beijing 100875 China department of mathematics University of Wisconsin-Milwaukee Milwaukee WI 53201 USA department of mathematics Central China Normal University Wuhan 430074 China

In this paper we study a certain directional Hilbert transform and the boundedness on some mixed norm spaces. As one of applications, we prove the Lp-boundedness of the Littlewood-Paley operators with variable kernels... 详细信息

关键词： directional Hilbert transform, Sobolev spaces, Littlewood-Paley operator,variable kernel.

L^p-Boundedness of Marcinkiewicz Integrals with Hardy Space Function Kernels

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Acta Mathematica Sinica,English Series 2000年第4期16卷 593-600页

作者： Yong Ding department of mathematics, Beijing Normal university, Beijing 100875, P. R. China Dashan Fan department of mathematics, Anhui university and university of wisconsin-Milwaukee. MW 53201 usa Yibiao Pan department of mathematics, university of Pittsburgh. Pittsburgh. Pennsylvania 15260. usa department of mathematics Beijing Normal University Beijing 100875 P. R. China department of mathematics Anhui University and University of Wisconsin-Milwaukee Wisconsin MW 53201 USA department of mathematics University of Pittsburgh Pittsburgh USA

We give the Lp-boundedness for a class of Marcinkiewicz integral operators μΩ, μΩ.λ and μΩ.S related to the Littlewood-Paley g-function, gλ-function and the area integral S, respectively. These operators have the kernel functions Ω∈H1(Sn-1), the Hardy space on Sn-1. The results in this paper substantially improve and extend the known results.

关键词： Marcinkiewicz integral g-function g_λ~*-function Area integral H^1 space Rough kernel

An OGCM Simulation of Seasonal and Interannual Variabilities in the Surface-Layer Pacific of the Equatorial Band

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Advances in Atmospheric Sciences 2002年第2期19卷 219-235页

作者： Boyin Huang and Zhengyu Liu department of Atmospheric and Oceanic Sciences, university of wisconsin-madison 1225 W. Dayton St., madison, wi-53.06, U.S.A. department of Earth Atmospheric and Planetary Sciences Massachusetts Institute of Technology Room 54-1721 77 Massachusetts Ave. 02139 Cambridge MA USA department of Atmospheric and Oceanic Sciences University of Wisconsin-Madison 1225 W. Dayton St. 53706 Madison WI USA

The heat budget is analyzed in the surface-layer (0-50 m) Pacific of the equatorial band (10°S-10°N), using the simulation of an ocean general circulation model from 1945 to 1993. The analysis indicates that downward net surface heat flux from the atmosphere and ocean advective heat fluxes play distinct roles in seasonal and interannual variabilities of surface-layer ocean temperature. The surface heat flux dominantly determines the ocean temperature in the seasonal time-scale. But, it has a negative feedback to the ocean temperature in the interannual time-scale. The interannual variability of ocean temperature is largely associated with the cold advection from off-equatorial divergent flow in the central Pacific and from upwelling in the cold tongue. Both the surface heat flux and ocean advective heat fluxes are important to the ocean temperature during an El Nino event. The ocean. advective heat fluxes are further associated with local westward trade wind in the central Pacific. These results are largely consistent with some regional observational analyses.

关键词： Seasonal variability Interannual variability Heat flux

Dedicated to *** Shoukun's 100th birthday

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Rare Metals 2006年第5期 391-392页

作者： Y.Austing Chang wisconsin Distinguished Professor department of Materials Science and Engineering university of wisconsin madison,wi 53.06 usa wisconsin Distinguished Professor department of Materials Science and Engineering university of wisconsin madison wi 53.06 usa

The pathway of the interdecadal variability in the Pacific Ocean

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Chinese Science Bulletin 2000年第17期45卷 1555-1561页

作者： WANG Dongxiao & LIU Zhengyu1. South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou 510301, China 2. department of Atmospheric & Oceanic Sciences, university of wisconsin-madison, madison, wi 53.06, usa Correspondence should be addressed to Wang Dongxiao South China Sea Institute of Oceanology Chinese Academy of Sciences Guangzhou China department of Atmospheric & Oceanic Sciences University of Wisconsin-Madison Madison USA

Cases of the interdecadal variability in the Pacific Ocean and their evolution were examined in the paper with the statistic methods (CEOF and composite analysis) over the period of 1950-1993. Observations of oceanic temperatures in the upper 400 m revealed an obvious region of the interdecadal signals in the central North Pacific. Such signals propagated south westward, then subducted to the subtropics. The hypothesized link for interdecadal oceanic variability between the subtropics and the tropics, especially with the western tropical Pacific was unraveled in order to detect the cause of decadal signals in the tropics. The thermal anomalies subducted in the central North Pacific east to the dateline only reach 18°N. There has been no further southward propagation since then due to a certain barrier. The origin of the interdecadal signals in the western tropical Pacific was traced to the southern tropical Pacific. There is a meridional pathway around the dateline where the signals were loaded.

关键词： interdecadal oceanic variability southern tropical Pacific signal pathway.

On Rectifiable Curves in the Plane

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Acta Mathematica Sinica,English Series 2001年第2期17卷 327-332页

作者： Shi Pan LU department of mathematics, Teachers College, Jimei university, Xiamen 361021, P. R. China Cheng Ming LEE department of Mathematical, Sciences, university of wisconsin-Milwaukee, Milwaukee, wisconsin 53201, usa department of mathematics Jimei University Xiamen 361021 P. R. China department of Mathematical Sciences University of Wisconsin-Milwaukee Milwaukee USA

This paper gives a formula of integral for the rectifiable curves in the plane by the Hausdorff fractional derivation and integral.

关键词： Rectifiable curve Hausdorff measure Hausdorff fractional derivation