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Shock Formation for 2D Isentropic Euler Equations with Self-similar Variables

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 349-412页

作者： Wenze SU school of mathematical sciences Fudan UniversityShanghai 200433China

The author studies the 2D isentropic Euler equations with the ideal gas *** exhibits a set of smooth initial data that give rise to shock formation at a single point near the planar *** solutions to the 2D isentropic Euler equations are associated with non-zero vorticity at the shock and have uniform-in-time-1/3-Holder ***,these point shocks are of self-similar type and share the same profile,which is a solution to the 2D self-similar Burgers *** proof of the solutions,following the 3D construction of Buckmaster,Shkoller and Vicol(in 2023),is based on the stable 2D self-similar Burgers profile and the modulation method.

关键词： 2D isentropic Euler equations Shock formation Self-similar solution

One-cycles on Gushel-Mukai fourfolds and the Beauville-Voisin filtration

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Science China Mathematics 2024年第4期67卷 713-732页

作者： Ruxuan Zhang school of mathematical sciences Fudan UniversityShanghai 200438China

We prove that the invariant locus of the involution associated with a general double EisenbudPopescu-Walter(EPW)sextic is a constant cycle surface and introduce a filtration on CH1of a Gushel-Mukai *** verify the sheaf/cycle correspondence for sheaves supported on low-degree rational curves,parallel to the case of cubic fourfolds of Shen-Yin’s work.

关键词： Beauville-Voisin ltration irreducible holomorphic symplectic variety Gushel-Mukai fourfold

Time-Periodic Solutions to Quasilinear Hyperbolic Systems on General Networks

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Chinese Annals of Mathematics,Series B 2024年第1期45卷 41-72页

作者： Peng QU school of mathematical sciences Fudan UniversityShanghai 200433China

For quasilinear hyperbolic systems on general networks with time-periodic boundary-interface conditions with a dissipative structure,the existence and stability of the time-periodic classical solutions are discussed.

关键词： Time-periodic solution Quasilinear hyperbolic system General net-work Classical solution Asymptotic stability

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

On the Bicanonical Map of Algebraic Threefolds of General Type

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Acta Mathematica Sinica,English Series 2024年

作者： Meng CHEN Chen JIANG Jian Shi YAN school of mathematical sciences & Shanghai Center for mathematical sciences Fudan University Shanghai Center for mathematical sciences & school of mathematical sciences Fudan University Department of Mathematics Northeastern University

We answer an open problem of the first author and Zhang(see Open Problem 6.4(3) in Math. Z. 258(2008), 565–585) and prove that, for any nonsingular projective 3-fold of general type with the geometric genus greater t...

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Coanalytic models for Hardy-type operators

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Science China Mathematics 2024年

作者： Xiangdi Fu Kunyu Guo Fugang Yan school of mathematical sciences Fudan University

We establish coanalytic models for a broad class of Hardy-type operators on L2[0, 1]. In particular,we show that the logarithmic Hardy operator is unitarily equivalent to the difference between the identity operator and the backward shift on a Bergman-type space. This result leads to several applications related to zero sets and invariant subspaces in weighted Bergman spaces. Additionally, we study logarithmic Hardy operators on Lp[0, 1] and obtain results concerning their boundedness, operator norms, and spectra.

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Ground state solutions to some indefinite nonlinear Schr?dinger equations on lattice graphs

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Acta Mathematica Sinica,English Series 2024年

作者： Wendi XU school of mathematical sciences Fudan University

In this paper, we consider the Schr?dinger type equation-?u+V (x)u=f(x, u) on the lattice graph ZNwith indefinite variational functional, where?is the discrete Laplacian. Specifically,we assume that V (x) and f(x, u) are periodic in x, f satisfies some growth condition and 0 lies in a finite spectral gap of (-?+V). We obtain ground state solutions by using the method of generalized Nehari manifold which has been introduced by Pankov.

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Existence of Global Solutions to the Nonlocal mKdV Equation on the Line

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Chinese Annals of Mathematics,Series B 2024年第4期45卷 497-528页

作者： Anran LIU Engui FAN school of mathematical sciences Fudan University

In this paper, the authors address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de Vries（nonlocal mKdV for short） equation under the initial data u0 ∈ H3（R） ∩H1,1（R） with the L1（R） small-norm assumption. A Lipschitz L2-bijection map between potential and reflection coefficient is established by using inverse scattering method based on a Riemann-Hilbert problem associated with the Cauchy problem. The map from initial potential to reflection coefficient is obtained in direct scattering transform. The inverse scattering transform goes back to the map from scattering coefficient to potential by applying the reconstruction formula and Cauchy integral operator. The bijective relation naturally yields the existence of global solutions in a Sobolev space H3（R） ∩ H1,1（R） to the Cauchy problem.

关键词： Global solutions Lipschitz continuous Nonlocal mKdV equation Plemelj projection operator Riemann-Hilbert problem

Weak type(1, 1) of the Riesz transform on some direct product manifolds with exponential volume growth

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Science China Mathematics 2024年第7期67卷 1599-1622页

作者： Hong-Quan Li Jie-Xiang Zhu school of mathematical sciences Fudan UniversityShanghai200433China Center for Applied Mathematics Tianjin UniversityTianjin300072China

In this paper, we are concerned with the Riesz transform on the direct product manifold H^(n)× M,where H^(n) is the n-dimensional real hyperbolic space, and M is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type(1, 1) property. In addition, we obtain the weak type(1, 1) of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type(1, 1) boundedness of some operators considered in the work of Li et al.(2016).

关键词： Riesz transform heat kernel direct product real hyperbolic space

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