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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

A UNIFORM CONVERGENT PETROV-GALERKIN METHOD FOR A CLASS OF TURNING POINT PROBLEMS

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Journal of Computational Mathematics 2024年第5期42卷 1356-1379页

作者： Li Feng Zhongyi Huang department of mathematical sciences Tsinghua UniversityBeijing 100084China

In this paper,we propose a numerical method for turning point problems in one dimension based on Petrov-Galerkin finite element method(PGFEM).We first give a priori estimate for the turning point problem with a single boundary turning *** we use PGFEM to solve it,where test functions are the solutions to piecewise approximate dual *** prove that our method has a first-order convergence rate in both L∞h norm and a discrete energy norm when we select the exact solutions to dual problems as test *** results show that our scheme is efficient for turning point problems with different types of singularities,and the convergency coincides with our theoretical results.

关键词： Turning point problem Petrov-Galerkin finite element method Uniform convergency

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

Jackson-type inequalities on high-dimensional spheres

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Science China Mathematics 2024年第11期67卷 2571-2586页

作者： Yeli Niu Feng Dai department of Mathematics Shanghai Normal University department of mathematical and Statistical sciences University of Alberta

We study Jackson's inequality on high-dimensional spheres with respect to the modulus of smoothness defined via the rotation *** obtain a version of Jackson's inequality with a dimensionfree constant,extending Newman and Shapiro's well-known results in 1964 from the case of r = 1 and p = ∞ to more general *** results partially overcome the curse of *** also establish similar results on the equivalence of the K-functional and modulus of smoothness.

关键词： modulus of smoothness $K$-functional Jackson's inequality high-dimensional approximation

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Equivariant Tautological Integrals on Flag Varieties

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 463-486页

作者： Xiaobo ZHUANG department of mathematical sciences Zhejiang Sci-Tech UniversityHangzhou 310018China

The author apples the Atiyah-Bott-Berline-Vergne formula to the equivariant tautological integrals over flag varieties of types A,B,C,D,and recovers the formulas expressing the integrals as iterated residues at infini... 详细信息

关键词： Localization Flag variety Residue

Integralinput-to-statestability ystemswithsmalldelays Integral input-to-state stability of systems with small delays

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Control Theory and Technology 2024年第3期22卷 455-467页

作者： R.H.Harsha Nawarathna Yuandan Lin Yuan Wang department of mathematical sciences Florida Atlantic UniversityBoca RatonFL 33431USA

We consider how small delays affect the integral-input-to-state stability(iss)property for a *** result is similar to the input-to-state stability(Iss)result obtained in[1]:the iss property will be preserved in a practical and semi-global manner if the delay interval is small ***,since the iss quantifies the robust stability in terms of a generalized Li norm of the inputs instead of a generalized Loo norm of the inputs for the Iss case,the techniques and proofs for the Iss case do not apply to the iss case *** the proofs in[1]are based on the Lyapunov-Razumikhin approach,our proofs are based on the iss-Lyapunov functions for the zero-delay *** addition to the interest by its own in showing how the iss property is affected by small delays,the result also serves to the study of the iss property for singularly perturbed systems.

关键词： Integral-input-to-state stability Nonlinear systems Small delays Lyapunov method

Emergent Geometry of KP Hierarchy

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Acta Mathematica Sinica,English Series 2024年第1期40卷 3-25页

作者： Jian ZHOU department of mathematical sciences Tsinghua UniversityBeijng 100084P.R.China

We explain how to construct a quantum deformation of a spectral curve associated to a tau-function of the KP *** construction is applied to Witten-Kontsevich tau-function to give a natural explanation of some earlier ... 详细信息

关键词： KP hierarchy tau-functions spectral curves