2nd Workshops on Mathematical Fluid Dynamics

Time：November 14-16, 2023

Venue：E4-233, Yungu Campus

ZOOM ID: 946 0347 2439

Passcode: 402724

I. Organizers

Alexey Cheskidov，Mimi Dai

II. Agenda

III. Talks

1. 9:00-10:00, November 14, Tuesday

Speaker: Zirong Zeng, Shanghai Jiao Tong University

Title: Sharp non-uniqueness of weak solutions to 3D hyperdissipative Navier-Stokes equations

Abstract: In this talk, I will present our recent work concerning the 3D hyperdissipative Navier-Stokes equations on the torus. It is well-known that, due to Lions, for any L^2 divergence -free initial data, there exist a unique smooth Leray-Hopf solution when viscosity exponent is larger than 5/4. We proved that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces in view of the LPS criteria. Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times.

2. 10:00-11:00, November 14, Tuesday

Speaker: Peng Qu, Fudan University

Title: Non-uniqueness for the hypo-viscous compressible Navier-Stokes equations

Abstract: We would discuss the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations in all dimensions d ≥ 2. For all hypo-viscosities (-Δ)^α with α ∈ (0, 1), we prove that there exist infinitely many weak solutions with the same initial data. This talk is based on a collaboration with Prof. Yachun Li, Prof. Deng Zhang and Dr. Zirong Zeng

3. 11:00-12:00, November 14, Tuesday

Speaker: Chunjing Xie, Shanghai Jiao Tong University

Title: The rigidity of steady solutions of high dimensional Navier-Stokes system and its applications

Abstract: The Liouville type theorem for stationary Navier-Stokes system in the whole space is longstanding open problem. In this talk, we first discuss the rigidity of steady Navier-Stokes system with dimension bigger than three in a class more general than self-similar solutions, where we do not need any type of self-similarity or smallness of solutions. Furthermore, this rigidity result is used to study the regularity and far field behavior of steady solutions of high dimensional Navier-Stokes system.

4. 10:00-11:00, November 15, Wednesday

Speaker: Deng Zhang, Shanghai Jiao Tong University

Title: Sharp non-uniqueness for the 3D MHD equations: beyond the Lions exponent

Abstract: In this talk I will present our results on the non-uniqueness of weak solutions to the 3D hyper viscous and resistive MHD equations in the space L_t^γ W_x^(s,p), where the exponents (s,γ,p) lie in two supercritical regimes. In particular, for the classical viscous and resistive MHD, the non-uniqueness is sharp near one endpoint of the Ladyzenskaja-Prodi-Serrin criterion. Moreover, the constructed weak solutions admit the spatial regularity outside a small fractal set of singular times with zero Hausdorff measure, where can be any given small positive constant. I will also show the strong vanishing viscosity and resistivity result, which yields the failure of Taylor’s conjecture along some sequence of weak solutions to the hyper viscous and resistive MHD beyond the Lions exponent. This talk is based on the joint work with Yachun Li and Zirong Zeng.

5. 11:00-12:00, November 15, Wednesday

Speaker: Yuanyuan Feng, East China Normal University

Title: Speeding up Langevin Dynamics by Mixing

Abstract: In this talk, we would study how stirring would help dissipate the energy and the applications, especially the application to the Langevin dynamics. We would first introduce the dissipation enhancing flows and study the dissipation time of such flows, where the dissipation time is explicitly computed based on the mixing rate. We then use mixing velocity field as a drift added to the Langevin dynamics (without changing the stationary distribution) and obtain quantitative estimates on the convergence rate of the system. We show that an exponentially mixing drift can be rescaled to make the mixing time of the Langevin system arbitrarily small.

6. 9:00-10:00, November 16, Thursday

Speaker: Jiajun Tong, Peking University

Title: Global Well-Posedness of the 2-D Peskin Problem under Geometric Conditions

Abstract: The 2-D Peskin problem describes coupled motion of a 1-D closed elastic string and the ambient Stokes flow in the plane. Its global well-posedness has been well-established when the initial string configuration is close to an equilibrium, which is an evenly-stretched circular configuration. In other words, initial shape of the string needs to be almost circular, and the string is almost evenly-stretched. In this talk, we present some recent progress on pursuing global solutions for a wider class of initial datum. We will show that certain geometric quantities of the string satisfy extremum principles and decay estimates. As a result, we can prove global well-posedness when the initial data satisfies a medium-size geometric condition on the string shape, while no assumption on the size of stretching is needed. This talk is based on a joint work with Dongyi Wei.

7. 10:00-11:00, November 16, Thursday

Speaker: Yu Feng, Peking University

Title: Mathematical problems in a class of tumor growth models

Abstract: In this talk, I will first introduce a class of tumor growth models described by porous medium equations (PME), which possesses a physical parameter m that characterizes the constitutive relation between cell density and pressure. And, one can obtain a Hele-Shaw type free boundary model by taking the incompressible limit (sending m to infinity). We then consider the forward and inverse problems in these tumor growth models. In the forward problem, we study the tumor boundary instability (the generation of finger-like structure) induced by nutrient consumption and supply based on the Hele-Shaw type model. While, in the inverse problem, we establish a Bayesian inversion framework for the PME models that perform uniformly well with respect to the constitutive relation parameter m.

8. 11:00-11:20, November 16, Thursday

Speaker: Jianing Yang, Shanghai Jiao Tong University

Title: Characterisations for the depletion of reactant in a one-dimensional dynamic combustion model

Abstract: In this talk, a novel observation is made on a one-dimensional compressible Navier—-Stokes model for the dynamic combustion of a reacting mixture of $\gamma$-law gases ($\gamma>1$) with discontinuous Arrhenius reaction rate function, on both bounded and unbounded domains. We show that the mass fraction of the reactant (denoted as Z) satisfies a weighted gradient estimate $Z_y/sqrt{Z}\in L_t^{\infty}L_y^2$, provided that at time zero the density is Lipschitz continuous and bounded strictly away from zero and infinity. Consequently, the graph of Z cannot form cusps or corners near the points where the reactant in the combustion process is completely depleted at any instant, and the entropy of Z is bounded from above.