召集人：李亚纯（上海交通大学）、梅茗（麦吉尔大学）、潘荣华（佐治亚理工学院）

时间：2024.4.21——4.27

会议日程 Talk Schedule

报告信息 Abstracts

1. Cheskidov, Alexey (Westlake University)

Dissipation anomaly for long time averages

Abstract: In turbulent flows, the energy injected at forced low modes (large scales) cascades to small scales through the inertial range where viscous effects are negligible, and only dissipates above Kolmogorov’s dissipation wavenumber. The persistence of the energy flux through the inertial range is what constitutes dissipation anomaly for viscous fluid flows as well as anomalous dissipation for the limiting inviscid flows. We first analyze these intrinsically linked phenomena on a finite time interval and prove the existence of various scenarios in the limit of vanishing viscosity, ranging from the total dissipation anomaly to a pathological one where anomalous dissipation occurs without dissipation anomaly, as well as the existence of infinitely many limiting solutions of the Euler equations in the limit of vanishing viscosity. Finally, expanding on the obtained total dissipation anomaly construction, we show the existence of dissipation anomaly for long time averages, relevant for turbulent flows, proving that the Doering-Foias upper bound is sharp.

2. Dai, Mimi (University of Illinois at Chicago)

Onsager conjecture for active scalar equations

Abstract: We will discuss recent development in construction of weak solutions for active scalar equations, for which the uniqueness and certain conservation law are violated. The purpose is to verify the sharp regularity threshold that separates the rigidity and flexibility regimes.

3. Duan, Renjun (The Chinese University of Hong Kong)

Landau damping of the two species Vlasov-Poisson system

Abstract: We present a result on nonlinear Landau damping for the two species Vlasov-Poisson system near Penrose stable equilibria on the torus T d × R d for arbitrary dimensions d ≥ 1. We 3extend the classical result of Mouhot and Villani for one species of electrons to two species in a physical situation where electrons mass is much smaller than ions mass. For the proof, we adopt the approach recently developed by Grenier, Nguyen and Rodnianski. Joint work with Zhiwen Zhang.

4. Hu, Xianpeng (The Hong Kong Polytechnic University)

Incompressible limit of compressible systems in R 3

Abstract: We will discuss the incompressible limit of compressible systems, including both compressible viscoelasticity and compressible Navier-Stokes equations. The ncompressible limit is characterised by the large value of the volume viscosity. On one hand, as the volume viscosity tends to infinity, the limit system of compressible viscoelasticity with smooth initial data is the incompressible viscoelasticity. On the other hand, as the volume viscosity tends to infinity, the limit system of compressible Navier-Stokes equations with discontinuous initial data is the density-dependent incompressible Navier-Stokes equation. For both settings, the global convergence to the limit system around an equilibrium is justified.

5. Jiu, Quansen (Capital Normal University)

Sharp decay estimates and asymptotic stability for incompressible MHD equations

without viscosity or magnetic diffusion

Abstract: In this talk, we consider decay estimates and asymptotic stability for n-dimensional incompressible MHD equations without viscosity or magnetic diffusion in a periodic domain, under initial perturbation around a background magnetic field satisfying the Diophantine condition. Through deeply exploring and effectively utilizing the structure of perturbation system, we discover a new dissipative mechanism, which enables us to establish the decay estimates and global well-posedness in the Sobolev spaces with lower regularity. Moreover, the decay estimates obtained can be seen as sharp in the sense that they are in line with those for the linearized system.

6. Li, Jinkai (South China Normal University)

Boundedness and unboundedness of entropy to the ideal gases in the presence of

far field vacuum

Abstract: As one of the basic physical states of the compressible gases, mathematical analysis of the entropy is an important issue in the studies of compressible flows. Due to the possible singularity of the entropy in the vacuum region, the dynamic behavior of the entropy, in particular the propagation of the uniform boundedness of the entropy, was rarely studied before. In this talk, we will present some recent studies on the boundedness and unboundedness of the entropy, in the presence of vacuum at the far field only. It will be shown that the boundedness or unboundedness of the entropy is determined by the algebraic decaying rate of the initial density at the far field. Precisely, the slow decaying rate implies the uniform boundedness locally or globally in time, while the fast decaying rate leads to the instantaneously blow up of the entropy.

7. Li, Jing (The Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

Global well-posedness of compressible Navier-Stokes equations with large data

Abstract: The barotropic compressible Navier-Stokes system subject to the Navier-slip boundary conditions in a general two-dimensional bounded simply connected domain is considered. For initial density allowed to vanish, the global existence of strong and weak solutions is established when the shear viscosity is a positive constant and the bulk one proportional to a power of the density with the power bigger than one and a third. It should be mentioned that this result is obtained without any restrictions on the size of initial value. To get over the difficulties brought by boundary, on the one hand, Riemann mapping theorem and the pull-back Green’s function method are applied to get a pointwise representation of the effective viscous flux. On the other hand, since the orthogonality is preserved under conformal mapping due to its preservation on the angle, the slip oundary conditions are used to reduce the integral¡br¿representation to the desired commutator form whose singularities can be cancelled out by using the estimates on the spatial gradient of the velocity. This is a joint work with Xinyu FAN and Jiaxu LI.

8. Liu, Shuangqian (Central China Normal University)

Couette flow to the Boltzmann equation in the hydrodynamic limit

Abstract: In this talk, I will report our recent study on the hydrodynamic limits of the Boltz

mann equation. Our investigation rigorously establishes the connection between incompressible Couette flow and the Boltzmann equation. This process involves a detailed derivation of Couette flow within the incompressible Navier-Stokes system from the Boltzmann equation. The core of our analysis relies on novel anisotropic estimates within the Wiener algebra function space.

9. Lv, Yong (Nanjing University)

Unconditional stability of equilibria in thermally driven compressible fluids

Abstract: We show that small perturbations of the spatially homogeneous equilibrium of a thermally driven compressible viscous fluid are globally stable. Specifically, any weak solution of the evolutionary Navier–Stokes–Fourier system driven by thermal convection converges to an equilibrium as time goes to infinity. The main difficulty to overcome is the fact the problem does not admit any obvious Lyapunov function. The result applies, in particular, to the Rayleigh–Benard convection problem.

10. Pan, Ronghua (Georgia Institute of Technology)

Isentropic approximation

Abstract: In the study of compressible flows, the isentropic model was often used to replace the more complicated full system when the entropy is near a constant. This is based on the expectation that the corresponding isentropic model is a good approximation to the full system when the entropy is sufficiently close to the constant. We will discuss the mathematical justification of isentropic approximation in Euler flows and in Navier-Stokes-Fourier flows.

11. Sheng, Wancheng (Shanghai University)

Simple waves and double waves of multi-dimensional hyperbolic equations for conservation laws

Abstract: In this talk, we are concerned with simple waves and double waves of flow patterns to multi-dimensional hyperbolic equations for conservation laws. The flow region of simple waves is covered by a family of single-parameter surfaces, on each of which flow is constant state. The double waves are a type of flow whose flow region is covered by a two parametric family of independent curves, along each of which flow is also constant state. The two-dimensional scaler hyperbolic equation for conservation law and the three-dimensional isentropic irrotational pseudo-steady flow are studied in some more detail.

12. Tan, Zhong (Xiamen University)

Mathematical modeling and theoretical exploration of magnetorheological fluid dampers

Abstract: A smart material - magnetorheological fluid: when no magnetic field is added, ferromagnetic particles are randomly suspended in a specific fluid, showing a low viscosity Newtonian fluid state; once magnetized, ferromagnetic particles are instantly magnetized into magnetic dipoles, connected by the head and tail into a columnar structure, and its viscosity rises tens of times, showing a non-Newtonian fluid state. At the same time, its shear stress would be enhanced so greatly that energy absorption and vibration reduction can be achieved. After the magnetic field is withdrawn, the fluid returns to its original state. In this talk, we will explain how this property can be used to provide theoretical support for the manufacture of new dampers for aircraft landing gear. According to the mathematical modeling skills, it is shown that the motion process can be modeled as a system of partial differential equations by applying mathematical ideas of conservation laws and careful calculus derivations. Therefore, the establishment of the partial differential equations and the basic theoretical problems involved in the collision and turbulence process, such as the stability problem, the free boundary problem and

the boundary layer problem, constitute the core technology of the manufacture of new aircraft damper.

13. Wang, Chunpeng (Jilin University, China)

Free boundary problems of degenerate elliptic equations and sonic jet flows from

convergent nozzles

Abstract: This talk concerns the compressible sonic jet flows from two-dimensional convergent nozzles with straight solid walls, which are governed by free boundary problems for a degenerate elliptic equation. The sonic jet problems are formulated and solved.

14. Wang, Qin (Yunnan University)

Some progress on two-dimensional Riemann problems and shock reflections

Abstract: In this talk, we will introduce some progress on two-dimensional Riemann problems and shock reflection. We will present a rigorous approach and related techniques to construct global solutions of the two-dimensional (2-D) Riemann problem for four-shock interactions for the Euler equations for potential flow. And we also introduce some new structures of shock regular reflections . This work is joint with Gui-Qiang G. Chen, Alex Cliffe, Feimin Huang, Song Liu and Junhe Zhou.

15. Wang, Yi (The Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

Time-asymptotic stability of basic wave patterns to compressible Navier-Stokes

equations

Abstract: First I will review the pioneer paper of Ilin-Oleinik for the time-asymptotic stability

of shock/rarefaction wave to Burgers equation in 1960 and the systematic studies for the timeasymptotic stability of individual wave pattern (shock, rarefaction wave or contact discontinuity) to one-dimensional (1D) system of the viscous conservation laws, e.g., compressilbe Navier-Stokes equations since 1980s. Then I will talk about the recent pregress on the time-asymptotic stability of generic Riemann solutions, consisting of multiple and different wave patterns, in particular, our recent developments on the time-asymptotic stability of the composite wave of viscous shock and rarefaction wave to 1D compressible isentropic Navier-Stokes equations and the superpositon of viscous shock, viscous contact wave and rarefaction wave to 1D full Navier-Stokes-Fourier equations. Last but not least, I will talk about the time-asymptotic stability of planar wave patterns to the multi-dimensional compressible Navier-Stokes equations and the stability of wave pattern to non-convex conservation laws.

16. Wen, Huanyao (South China University of Technology)

Some recent progress on blowup criteria for compressible Navier-Stokes equations

Abstract: In this talk, we will introduce some recent progress on blowup criteria for viscous, compressible, and heat-conducting flow with vaccum.

17. Xiang, Wei (The City University of Hong Kong)

Convexity, uniqueness, and stability of the regular shock reflection-diffraction prob

lem

Abstract: We will talk about our recent results on the regular reflection solutions for the potential flow equation in a natural class of self-similar solutions. The approach is based on a nonlinear version of the method of continuity and the renormalization to a transport equation.

18. Yang, Xiaozhou (The Innovation Academy for Precision Measurement Science and Technology

(APM), Chinese Academy of Sciences)

Triple shock wave structure of two-dimensional hyperbolic systems of conservation

laws and related problems

Abstract: In this talk, we will discuss the three-pronged shock structures formed between nonself-similar shock waves of a class of two-dimensional hyperbolic systems of conservation laws, the structures of which are relatively rare. The equations furthermore contain two-dimensional non-self-similar contact discontinuity waves. All of the waves above, including the two-dimensional intermediate states, can be expressed by the implicit functions. This work is related to work with Professor Chen Gui-Qiang and Professor Wang Dehua.

19. Zhang, Deng (Shanghai Jiao Tong University)

Non-uniqueness in law of Leray solutions to 3D forced stochastic Navier-Stokes

equations

Abstract: This talk concerns the forced stochastic Navier-Stokes equation driven by additive noise in the three dimensional Euclidean space. By constructing an appropriate forcing term, we show that there exist distinct Leray solutions in the probabilistically weak sense. In particular, the joint uniqueness in law fails in the Leray class. The non-uniqueness also displays in the probabilistically strong sense in the local time regime, up to stopping times. Furthermore, we discuss the optimality from two different perspectives: sharpness of the hyper-viscous exponent and size of the external force. As a consequence, one derives that the Lions exponent is the sharp viscosity threshold for the niqueness/non-uniqueness in law of Leray solutions.

20. Zhang, Yinghui (Guangxi Normal University)

Global well–posedness and large time behavior of classical solutions to a generic

compressible two–fluid model

Abstract: We investigate a generic compressible two–fluid model with common pressure (P ⁺ = P ⁻) in R ³ . The global well–posedness of the 3D compressible two–fluid model with common pressure has been a challenging open problem due to the fact that the system is partially dissipative and its nonlinear structure is very terrible. In the present work, by exploiting the dissipation structure of the model and making full use of several key observations, we prove global existence and large time behavior of classical solutions to the 3D compressible two–fluid model with common pressure. The method relies upon careful analysis of the linearized system, exploitation of the algebraic structure of the nonlinear system, and the introduction of an auxiliary velocity v = (2µ ⁺ + λ ⁺)u ⁺ − (2µ ⁻ + λ ⁻)u ⁻ which plays the role of the effective viscous flux (since in this system P ⁺ = P ⁻) in the single phase case: such velocity has better regularity than phase velocities u ^±. 

参会人员 Participants

1. Cao, Gaowei 曹高伟 (The Innovation Academy for Precision Measurement Science and Technol

ogy (APM), Chinese Academy of Sciences, China)

2. Chen, Jiahuan 陈嘉欢 (Shanghai Jiao Tong University, China)

3. Cheng, Hongjun 程红军 (Yunnan University, China)

4. Cheskidov, Alexey (Westlake University, China)

5. Dai, Mimi 代觅秘 (University of Illinois at Chicago, USA)

6. Ding, Min 丁敏 (Wuhan University of Technology, China)

7. Du, Runmei 杜润梅 (Changchun University of Technology, China)

8. Duan, Renjun 段仁军 (The Chinese University of Hong Kong, China)

9. Fan, Lili 范丽丽 (Wuhan Polytechnic University, China)

10. Gao, Pu 高普 (Shanghai University, China)

11. Geng, Shifeng 耿世锋 (Xiangtan University, China)

12. Han, Xiaomin 韩小敏 (Nanchang University, China)

13. Hu, Xianpeng 胡先鹏 (The Hong Kong Polytechnic University, China)

14. Jin, Rui 金锐 (Shanghai Jiao Tong University, China)

15. Jiu, Quansen 酒全森 (Capital Normal University, China)

16. Kan, Hui 阚辉 (The Innovation Academy for Precision Measurement Science and Technology

(APM), Chinese Academy of Sciences, China)

17. Li, Haitong 黎海彤 (Changchun University of Technology, China)

18. Li, Hao 李浩 (Zhejiang Normal University, China)

19. Li, Jinkai 李进开 (South China Normal University, China)

120. Li, Jing 李竞 (The Academy of Mathematics and Systems Science, Chinese Academy of Sciences,

China)

21. Li, Lixin 李立馨 (Guangxi Normal University, China)

22. Li, Yin 李银 (Guangdong University of Finance and Economics, China)

23. Liao, Rujin 廖如进 (Nanchang University, China)

24. Liu, Cunming 刘存明 (Qufu Normal University, China)

25. Liu, Jinjing 刘进静 (Yunnan University, China)

26. Liu, Lihui 刘力珲 (Shanghai Jiao Tong University, China)

27. Liu, Manyu 刘漫雨 (Nanchang University, China)

28. Liu, Shuangqian 刘双乾 (Central China Normal University, China)

29. Lv, Yong 吕勇 (Nanjing University, China)

30. Mai, Lasu 麦拉苏 (Inner Mongolia University, China)

31. Qin, Guoquan 秦国权 (Yunnan University, China)

32. Shang, Zhangyang 尚朝阳 (Shanghai Lixin University of Accounting and Finance, China)

33. Sheng, Wancheng 盛万成 (Shanghai University, China)

34. Song, Haoxiang 宋昊翔 (Shanghai University, China)

35. Tan, Zhong 谭忠 (Xiamen University, China)

36. Tong, Leilei 童雷雷(Chongqing University of Posts and Telecommunications)

37. Wang, Chunpeng 王春朋 (Jilin University, China)

38. Wang, Hongyu 王弘宇 (Nanchang University, China)

39. Wang, Huaqiao 王华桥 (Chongqing University, China)

40. Wang, Qin 王钦 (Yunnan University, China)

41. Wang, Tingting 王婷婷 (Shanghai University, China)

42. Wang, Yi 王益 (The Academy of Mathematics and Systems Science, Chinese Academy of Sci

ences, China)

43. Wen, Huanyao 温焕尧 (South China University of Technology, China)

44. Wu, Guochun 吴国春 (Huaqiao University, China)

45. Xi, Shuai 袭帅 (Shandong University of Science and Technology, China)

46. Xiang, Wei 向伟 (The City University of Hong Kong, China)

47. Xiao, Qinghua 肖清华 (The Innovation Academy for Precision Measurement Science and Tech

nology (APM), Chinese Academy of Sciences, China)

48. Xu, Jianing 徐佳宁 (Jilin University, China)

49. Yang, Xiaozhou 杨小舟 (The Innovation Academy for Precision Measurement Science and Tech

nology (APM), Chinese Academy of Sciences, China)

50. Yang, Yueying 杨月颖 (Shanghai University, China)

51. Yuan, Bing 袁冰 (Nanchang University, China)

52. Zeng, Zirong 增梓 (Shanghai Jiao Tong University, China)

53. Zhang, Lizhen 张丽珍 (Shanghai Jiao Tong University, China)

54. Zhang, Deng 张登 (Shanghai Jiao Tong University, China)

255. Zhang, Jian 张健 (The Academy of Mathematics and Systems Science, Chinese Academy of

Sciences, China)

56. Zhang, Yinghui 张映辉 (Guangxi Normal University, China)

57. Zheng, Zhiwei 郑志伟 (Nanjing University, China)

58. Zhao, Liang 赵亮 (Oxford Suzhou Center for Advanced Research, China)

59. Zhang, Zhipeng 张志朋 (Ocean University of China, China)