郝成春 - 个人网站

摘要: This talk is mainly concerned with the complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary. Uniquely, we make NO assumptions on symmetry, periodicity, graph representation, or domain topology (simple connectivity). At the maximal existence time $T<\infty$, up to which the velocity field and the free boundary can be continued in $H^3\times H^4$, blow-up must occur in at least one of five mutually exclusive ways: (i) self-intersection of the free boundary for the first time; (ii) loss of mean curvature regularity in $H^{\frac{3}{2}}$, or the free boundary regularity in $H^{2+\varepsilon}$ (for any sufficiently small constant $\varepsilon>0$); (iii) loss of $H^{\frac{5}{2}}$ regularity for the normal boundary velocity; (iv) the $L^1_tL^\infty$-blow-up of the tangential velocity gradient on the boundary; or (v) the $L^1_tL^\infty$-blow-up of the full velocity gradient in the interior. Furthermore, for simply connected domains, blow-up scenario (v) simplifies to a vorticity-based Beale-Kato-Majda criterion, and in particular, irrotational flows admit blow-up only at the free boundary. This talk is based on the joint work (arXiv: 2507.10032) with Tao Luo and Siqi Yang.

摘要: This talk addresses the free boundary problem for an approximate model describing a gas bubble of finite mass within a bounded incompressible viscous liquid. The model incorporates surface tensions at both the gas-liquid interface and the external free surface of the entire gas-liquid region. It is shown that any regular, spherically symmetric steady-state solution is characterized by a positive root of a ninth-degree polynomial. We prove the existence and uniqueness of these roots, and establish a one-to-one correspondence between equilibria and pairs of gas mass and liquid volume.
We demonstrate that these equilibria exhibit both nonlinear and exponential asymptotic stability under small perturbations that preserve gas mass and liquid volume. Moreover, an equilibrium solution serves as a local minimizer of the energy functional, even under relatively large perturbations, with the proportionality constant determined by the adiabatic constant. Additionally, we construct a global center manifold and apply center manifold theory to study the dynamics of the system.
Our findings apply to gases and liquids of all sizes. We also derive the optimal exponential decay rate for small liquid volumes by analyzing the spectrum bounds of the associated linear operator. Our results indicate that decreasing the gas mass or increasing the temperature can accelerate the convergence rate—this behavior is distinct from unbounded liquid scenarios.
This talk is based on joint research with Tao Luo and Siqi Yang (arXiv:2408.15587).

摘要: This talk is mainly concerned with the free boundary problem for an approximate model (for example, arising from the study of sonoluminescence) of a gas bubble of finite mass enclosed within a bounded incompressible viscous liquid, accounting for surface tensions at both the gas-liquid interface and the external free surface of the entire gas-liquid region. It is found that any regular spherically symmetric steady-state solution is characterized by a positive root of a ninth-degree polynomial for which the existence and uniqueness are proved and a one-to-one correspondence between equilibria and pairs of gas mass and liquid volume is established. We prove that these equilibria exhibit nonlinear and exponential asymptotic stability under small perturbations that conserve gas mass and liquid volume, and an equilibrium solution acts as a local minimizer of the energy functional, even under relatively large perturbations, with the proportionality constant determined by the adiabatic constant. Moreover, we construct a global center manifold to apply the center manifold theory. Our results apply to gases and liquids of all sizes. Furthermore, we derive the optimal exponential decay rate for small liquid volumes by analyzing the spectrum bounds of the associated linear operator and show that decreasing the gas mass or increasing the temperature can accelerate the convergence rate, a behavior not seen in unbounded liquid scenarios. This talk is based on the joint work (arXiv:2408.15587) with Tao Luo and Siqi Yang.

摘要: This talk is mainly concerned with the free boundary problem for an approximate model (for example, arising from the study of sonoluminescence) of a gas bubble of finite mass enclosed within a bounded incompressible viscous liquid, accounting for surface tensions at both the gas-liquid interface and the external free surface of the entire gas-liquid region. It is found that any regular spherically symmetric steady-state solution is characterized by a positive root of a ninth-degree polynomial for which the existence and uniqueness are proved and a one-to-one correspondence between equilibria and pairs of gas mass and liquid volume is established. We prove that these equilibria exhibit nonlinear and exponential asymptotic stability under small perturbations that conserve gas mass and liquid volume, and an equilibrium solution acts as a local minimizer of the energy functional, even under relatively large perturbations, with the proportionality constant determined by the adiabatic constant. Moreover, we construct a global center manifold to apply the center manifold theory. Our results apply to gases and liquids of all sizes. Furthermore, we derive the optimal exponential decay rate for small liquid volumes by analyzing the spectrum bounds of the associated linear operator and show that decreasing the gas mass or increasing the temperature can accelerate the convergence rate, a behavior not seen in unbounded liquid scenarios. This talk is based on the joint work (arXiv:2408.15587) with Tao Luo and Siqi Yang.

摘要: I will survey some recent results concerning the free boundary problem of incompressible MHD in a bounded domain with a closed surface. Specifically, I will focus on the main ideas presented in the a priori estimates for cases involving surface tension.

摘要: In this presentation, I will provide an overview of the local regularity theory concerning the three-dimensional free boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension. The focus will be on understanding how the geometric properties of the free boundary influence the regularity of smooth solutions, under certain conditions. These conditions encompass the low regularity of the free boundary (including curvature and normal velocity constraints), as well as conditions on the fluid velocity and magnetic fields to prevent self-intersection of the free boundary. This analysis applies generally to bounded domains with closed surfaces, without assuming the free boundary to be a graph. The regularity estimates are primarily influenced by bounds on curvature and pressure, which are linked to the initial mean curvature. Notably, the initial velocity on the boundary is not a critical factor for the final regularity estimate. Furthermore, this approach avoids addressing the spatial regularity of the flow map in Lagrangian coordinates, which can be limited in terms of maximal regularity and the geometric characteristics of the evolving domain. This talk is based on joint research with Siqi Yang [arXiv:2312.09473], where we delve into these intricate relationships and their implications in the context of MHD equations with surface tension.

摘要: In this talk, we are talking about the existence of smooth initial data for the 2d free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity (self-intersects in at least one point) in finite time. It turns out that the presence of the magnetic field does not prevent the viscous fluid to form splash singularities for certain smooth initial data. This talk is based on the joint work with Siqi Yang.

摘要: 针对不可压缩理想磁流体方程自由边界问题的适定性，对有界初始区域的情形做较详细的阐述，并对相关的背景、研究进展和存在的问题作简要的介绍。