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On asymptotic stability of the 3D Boussinesq equations without heat conduction

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Science China Mathematics 2024年第2期67卷 253-280页

作者： Lihua Dong Yongzhong Sun School of physical and mathematical sciences Nanjing Tech UniversityNanjing 211816China department of Mathematics Nanjing UniversityNanjing 210093China

We investigate the asymptotic stability of the solutions to Boussinesq equations without heat conduction with the initial data near a specific stationary solution in the three-dimensional domain Ω=R^(2)×(0,1).It is shown that the solution starting from a small perturbation to the stationary solution converges to it with explicit algebraic rates as time tends to infinity.

关键词： Boussinesq equations stationary solutions asymptotic stability decay rates

NSNO:Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium

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Journal of Systems Science & Complexity 2024年第2期37卷 413-440页

作者： CHEN Fukai LIU Ziyang LIN Guochang CHEN Junqing SHI Zuoqiang department of mathematical sciences Tsinghua UniversityBeijing 100084China Yau mathematical sciences Center Tsinghua UniversityBeijing 100084China Yanqi Lake Beijing Institute of mathematical sciences and Applications Beijing 101408China

In this paper,the authors propose Neumann series neural operator(NSNO)to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to *** equation is a crucial partial differential equation(PDE)with applications in various scientific and engineering ***,efficient solver of Helmholtz equation is still a big challenge especially in the case of high ***,deep learning has shown great potential in solving PDEs especially in learning solution *** by Neumann series in Helmholtz equation,the authors design a novel network architecture in which U-Net is embedded inside to capture the multiscale *** experiments show that the proposed NSNO significantly outperforms the state-of-the-art FNO with at least 60%lower relative L^(2)-error,especially in the large wavenumber case,and has 50%lower computational cost and less data ***,NSNO can be used as the surrogate model in inverse scattering *** tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.

关键词： Helmholtz equation inverse problem neumann series neural network solution operator

Spin-based magnetic random-access memory for high-performance computing

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National Science Review 2024年第3期11卷 20-23页

作者： Kaiming Cai Tianli Jin Wen Siang Lew School of Physics Huazhong University of Science and Technology Interuniversity Microelectronics Centre(IMEC) School of physical and mathematical sciences Nanyang Technological University

Memory serves as a critical component in today's electronic systems for data storage and processing. In traditional computer architectures, the logic and memory units are physically separated, due to the performance gap in operational speed and capacity among memories, resulting in the fundamental limitation of the von Neumann computers.

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GLOBAL SOLUTIONS IN THE CRITICAL SOBOLEV SPACE FOR THE LANDAU EQUATION

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Acta Mathematica Scientia 2024年第4期44卷 1347-1372页

作者： Hao WANG department of mathematical sciences Tsinghua UniversityBeijing 100084China

The Landau equation is studied for hard potential with-2≤γ≤*** a perturbation setting,a unique global solution of the Cauchy problem to the Landau equation is established in a critical Sobolev space H_(x)^(d)L_(v)^(2)(d>3/2),which extends the results of[11]in the torus domain to the whole space R_(x)^(3).Here we utilize the pseudo-differential calculus to derive our desired result.

关键词： Landau equation nonlinear energy method global existence pseudo-differential calculus

Existence and asymptotics of normalized solutions for the logarithmic Schrödinger system

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Science China Mathematics 2024年第9期67卷 2019-2048页

作者： Qian Zhang Wenming Zou department of mathematical sciences Tsinghua UniversityBeijing 100084China

This paper is concerned with the following logarithmic Schrodinger system:{-Δu_(1)+ω_(1)u_(1)=u_(1)u_(1)logu_(1)^(2)+2p/p+q|u_(2)|^(q)|u_(1)|^(p-2)u_(1),-Δu_(2)+ω_(2)u_(2)=u_(2)u_(2)log u_(2)^(2)+2q/p+q|u_(1)|^(p)|u_(2)|^(q-2)u_(2),∫_(Ω)|u_(i)|^(2)dx=ρ_(i),i=1,2,(u_(1),u_(2))∈H_(0)^(1)(Ω;R^(2)),where Ω=R^(N)or Ω■R^(N)(N≥3)is a bounded smooth domain,andω_(i)R,μ_(i),ρ_(i)>0 for i=1,***,p,q≥1,and 2≤p+q≤2^(*),where 2^(*):=2N/*** using a Gagliardo-Nirenberg inequality and a careful estimation of u log u^(2),firstly,we provide a unified proof of the existence of the normalized ground state solution for all 2≤p+q≤2^(*).Secondly,we consider the stability of normalized ground state ***,we analyze the behavior of solutions for the Sobolev-subcritical case and pass to the limit as the exponent p+q approaches 2^(*).Notably,the uncertainty of the sign of u log u^(2)in(0,+∞)is one of the difficulties of this paper,and also one of the motivations we are interested *** particular,we can establish the existence of positive normalized ground state solutions for the Brézis-Nirenberg type problem with logarithmic perturbations(i.e.,p+q=2^(*)).In addition,our study includes proving the existence of solutions to the logarithmic type Bréis-Nirenberg problem with and without the L^(2)-*** ∫_(Ω)|u_(i)|^(2)dx=ρ_(i)(i=1,2)by two different methods,*** results seem to be the first result of the normalized solution of the coupled nonlinear Schrodinger system with logarithmic perturbations.

关键词： logarithmic Schrodinger system Brézis-Nirenberg problem normalized solution existence and stability behavior of solutions

Noncommutative analysis of Hermite expansions

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Science China Mathematics 2024年第10期67卷 2331-2356页

作者： Bang Xu department of Mathematics University of Houston department of mathematical sciences Seoul National University

This paper is devoted to the study of semi-commutative harmonic analysis associated with Hermite semigroups. In the first part, we establish the noncommutative maximal inequalities for Bochner-Riesz means associated with Hermite operators and then obtain the corresponding pointwise convergence theorems. In particular, we develop a noncommutative version of Stein's theorem of Bochner-Riesz means for Hermite operators. In the second part, we investigate two noncommutative multiplier theorems. Our approach in this part relies on a noncommutative analog of the classical Littlewood-Paley-Stein theory associated with Hermite semigroups.

关键词： noncommutative $L_{p}$-spaces Hermite semigroup maximal inequalities pointwise convergence multipliers

Markovian Quadratic BSDEs with an Unbounded Sub-quadratic Growth

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 441-462页

作者： Jingnan JU Shanjian TANG Shanghai Center for mathematical sciences Fudan UniversityShanghai 200433China department of Finance and Control sciences School of Mathematical SciencesFudan UniversityShanghai 200433China

This paper is devoted to the solvability of Markovian quadratic backward stochastic differential equations(BSDEs for short)with bounded terminal *** generator is allowed to have an unbounded sub-quadratic growth in the second unknown variable *** existence and uniqueness results are given to these *** an application,an existence result is given to a system of coupled forward-backward stochastic differential equations with measurable coefficients.

关键词： Markovian BSDE Quadratic growth Unbounded sub-quadratic term coeficients Coupled FBSDE

Pseudo-Differential Operators of Homogeneous Symbol Class Associated with the Weinstein Transform

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Acta Mathematica Sinica,English Series 2024年第6期40卷 1533-1562页

作者： Santosh Kumar UPADHYAY Mohd SARTAJ department of mathematical sciences Indian Institute of Technology(BHU)VaranasiIndia

In this paper,pseudo-differential operators with homogeneous symbol classes associated with the Weinstein transform are *** boundedness of pseudo-differential operators and commutator between two pseudo-differential o... 详细信息

关键词： Pseudo-Differential Operators Homogeneous Symbol Class Weinstein Operator Sobolev space Weinstein transform

On the existence and stability of traversable wormhole solutions with novel shapefunctions in the framework of F(R,L_(m))gravity

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Communications in Theoretical Physics 2024年第5期76卷 114-127页

作者： Sourav Chaudhary Jitendra Kumar S K Maurya Sweeti Kiroriwal Abdul Aziz department of Mathematics Central University of HaryanaJant-PaliMahendergarhIndia department of mathematical and physical sciences College of Arts and ScienceUniversity of NizwaP.O.Box 33Nizwa 616Oman

In this work,we have explored wormhole(WH)solutions in F(R,L_(m))gravity by assuming the Morris-Thorne WH metric and F(R,Lm)=R/2+(1+γR)L_(m),whereγis the free model *** determined the WH solutions by utilizing two newly developed shape functions(SF)that satisfy all basic conditions for a WH’s physical *** also observe that the null energy condition(NEC)behaves ***,for both models,we use the volume integral quantifier(VIQ)and Tolman-Oppenheimer-Volkoff(TOV)equation to determine how much exotic matter is needed near the WH throat and the stability of the *** extensive detailed discussions of the matter components have been done via graphical *** obtained WH geometries meet the physically acceptable conditions for a stable wormhole.

关键词： energy conditions wormhole exotic matter F(R L_(m))gravity