The second workshop brings together researchers working on the mathematical analysis of partial differential equations arising in fluid dynamics. It provides a forum for presenting recent developments, exchanging ideas, and discussing challenging open problems.
The programme encourages interaction between established researchers and young mathematicians, spanning incompressible and compressible flows, kinetic equations, hyperbolic control systems, free-boundary problems, regularity, and flexibility.
Abstracts are listed in programme order. Open a title to read the full abstract. All 11 talk titles and abstracts are now available.
01Time dependent Prandtl Ansatz always fails in Sobolev spacesDongfen Bian · Beijing Institute of TechnologyOctober 22 · 09:00–10:00 · Chair: Quoc-Hung Nguyen
The aim of this talk is to prove that Prandtl Ansatz, which describes the boundary layer which appears in an incompressible fluid as the viscosity goes to 0, is always false in Sobolev spaces.
02Uniform Estimates for the Landau–Fermi–Dirac EquationYuzhe Zhu · The Chinese University of Hong Kong, ShenzhenOctober 22 · 10:30–11:30 · Chair: Zhilong Xue
We will discuss the Landau–Fermi–Dirac equation and uniform regularity estimates for its solutions. The estimates are uniform with respect to the quantum parameter and are useful for understanding the classical limit.
03Maximal Regularity and Caloric Trace Estimates in Mixed Lebesgue Norms for the Heat EquationHui Chen · Zhejiang University of Science and TechnologyOctober 22 · 11:30–12:30 · Chair: Zhilong Xue
We study the heat equation in the half-space with nonhomogeneous Dirichlet boundary data. For the caloric extensionof the boundary data, we prove maximal regularity estimates in mixed Lebesgue normsfor any order derivative ofin terms of mixed Besov and Lizorkin–Triebel type norms of. We also establish the corresponding reverse inequalities, which are caloric trace estimates recovering the boundary regularity offrom the mixed-norm regularity of. In particular, the endpoint casesorin the maximal regularity estimate are included and appear to be new and may be useful in the analysis of free-boundary Navier–Stokes problems with small initial data, whereas the caloric trace estimates may be relevant to the construction of Stokes or Navier–Stokes flows exhibiting strong boundary singularities.
04Local Well-Posedness for Compressible Water Waves with Acute Contact AnglesChao Wang · Peking UniversityOctober 22 · 14:00–15:00 · Chair: Quoc-Hung Nguyen
In this talk, we will talk about the local well-posedness of the compressible Euler equations in a two-dimensional bounded corner domain with acute contact angles. This configuration describes a free surface intersecting the fixed bottom at two points, where the fluid is subject to a gravitational field and the interface between the fluid and air is influenced by capillary forces, without assuming irrotationality. When the contact angles are less than, the local existence theory is established for the solution, with dissipation effects occurring at the contact points. The main analytical challenge arises from the contact points, which renders previous methods for dealing with compressible free boundary problems inadequate. To overcome this, we first establish the geometric structure for the compressible Euler equations, an approach originally introduced by Shatah and Zeng for incompressible fluids. Additionally, we provide a singularity analysis for the wave equations in the corner domain, which ensures the validity of calculations near the corner. Finally, based on the geometric structure and singularity analysis, we obtain a priori energy estimates. Using these estimates, we also prove the local well-posedness of the solutions in a geometric formulation. To our knowledge, this is the first result addressing compressible Euler equations with a free boundary involving contact points.
05Climbing theLadder in the Euler Equations and Isometric EmbeddingsWentao Cao · Capital Normal UniversityOctober 22 · 15:30–16:30 · Chair: Hui Chen
In this talk, I will present the history and the approaches to addressing Onsager's conjecture forweak solutions of the incompressible Euler equations that violate the energy equality, as well as those for establishing the flexibility ofisometric embeddings.
06Global-in-Time Inviscid Limit of the Navier–Stokes Equations for Vortex–Wave DataTrinh Nguyen · Shanghai Jiao Tong UniversityOctober 22 · 16:30–17:30 · Chair: Zhilong Xue
We justify the vortex–wave system introduced by Marchioro and Pulvirenti as the vanishing-viscosity limit of the two-dimensional Navier–Stokes equations on arbitrary finite time intervals. The initial vorticity consists of a point vortex and a smooth, compactly supported background that vanishes near the vortex. The proof relies on two key ingredients. First, we track the transported support of the approximate background to maintain separation from the point vortex, removing the restriction involving the region swept out by past wave supports. Second, we use Gallay's weighted energy method to cancel the leading ambient-strain contribution. The regular component converges strongly to the background vorticity in, while the concentrated component converges to a Lamb–Oseen vortex centered on the limiting point-vortex trajectory. We obtain quantitative estimates for both components, with a viscosity threshold depending on the chosen time interval, thus establishing the global-in-time validity of the vortex–wave system.
In this talk, I will review results on the null-controllability, exact controllability of hyperbolic systems in one dimensional space for the optimal time. Both time-independent and time-dependent coefficients are discussed. Interesting and surprising phenomena occur. There is a finite gap for the optimal time in these cases even the time is involved only in the zero-order coefficients and even these coefficients are infinitely smooth. When the analyticity in time of the coefficients enters, the gap disappears. This is based on joint work with Jean-Michel Coron.
08Necessary and Sufficient Conditions for Exact Controllability of Linear Hyperbolic Systems inSpaces,Yacine Chitour · Université Paris-SaclayOctober 23 · 10:30–11:30 · Chair: Zhilong Xue
In this talk I will present necessary and sufficient conditions of Hautus type for the approximate and exact controllability of linear hyperbolic systems and linear controlled difference equations inspaces,. The argument for the caserelies on realization theory and reduces to a corona problem while the argument for the caseuses an observability criterion introduced by Russell and Weiss. This is joint work with S. Fueyo, P. Lissy, G. Mazanti and M. Sigalotti.
09Local Well-Posedness for the Boltzmann Equation with Hard PotentialsWei-Xi Li · Wuhan UniversityOctober 23 · 11:30–12:30 · Chair: Yuzhe Zhu
We consider the spatially inhomogeneous non-cutoff Boltzmann equation with hard potentials in the non-perturbative setting. For initial data with polynomial decay in the velocity variable, we establish the local-in-time existence and uniqueness of weak solutions, conditional to pointwise bounds on the hydrodynamic quantities (mass, energy, and entropy). Compared to the soft potential case, the key challenge for full-range hard potentials lies in the more severe loss of velocity moments. The proof combines a hypoelliptic estimate with interpolation inequalities to handle the moment-loss terms.
10The Boltzmann Equation in the Homogeneous Critical Regularity FrameworkJiang Xu · Nanjing University of Aeronautics and AstronauticsOctober 23 · 14:00–15:00 · Chair: Quoc-Hung Nguyen
We construct a unique global solution with critical regularity to the Cauchy problem for the 3D Boltzmann equation with initial data near the Maxwellian in the spatially critical Besov space. Furthermore, under the additional assumption that the low-frequency part of the initial perturbation is bounded inwith, it is shown that the solution converges to its equilibrium in large times with the optimal rate ofinwith some, and the microscopic part decays at an enhanced rate of. This is the first work addressing the global existence and large-time behavior of the Boltzmann equation in the homogeneous critical setting, which relies on a new understanding of hypocoercivity theory indicating Lyapunov functionals with different dissipation rates in low and high frequencies. Moreover, the time-weighted Lyapunov energy argument has been developed to get the optimal time-decay estimates.
11Nonuniqueness for the SQG Equation with anActive ScalarRui Jin · Shanghai Jiao Tong UniversityOctober 23 · 15:30–16:30 · Chair: Zhilong Xue
We prove a flexibility theorem for weak solutions of the inviscid surface quasi-geostrophic equation on the two-dimensional torus, in the standard momentum formulation, with active scalarfor. We also establish nonuniqueness for every initial datum in a dense subset of the mean-zero space. The proof uses a convex-integration scheme based on concentrated traveling SQG profiles and a bilinear null-form estimate.
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05 · Resources
Conference materials
会议资料
PDF · Programme & abstracts
Conference handbook
The complete programme with confirmed session chairs, all 11 abstracts, dining arrangements and a QR code linking to this conference webpage.
Arrival: October 21 · Departure: October 24 Buffet lunches: October 22 & 23, Wuke Hotel Restaurant Conference dinner: October 22 evening, Wuke Hotel Restaurant (time TBA) Accommodation arrangements: TBA
