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Chengchun Hao

Publications and source records attributed to Chengchun Hao.

At least 19 recordsLinked to original sources
arXiv2025

Finite-Time Splash in Free Boundary Problem of 3D Neo-Hookean Elastodynamics

This paper establishes finite-time splash singularity formation for 3D viscous incompressible neo-Hookean elastodynamics with free boundaries. The system features mixed stress-kinematic conditions where viscous-elastic stresses balance pressure forces at the evolving interface -- a configuration generating complex boundary integrals that distinguish it from Navier-Stokes or MHD systems. To address this challenge, we employ a Lagrangian framework inspired by Coutand and Shkoller (2019), developing specialized coordinate charts and constructing a sequence of shrinking initial domains with cylindrical necks connecting hemispherical regions to bases. Divergence-free initial velocity and deformation tensor fields are designed to satisfy exact mechanical compatibility. Uniform a priori estimates across the domain sequence demonstrate that interface evolution preserves local smoothness while developing finite-time self-intersection. Energy conservation provides foundational stability, while higher-order energy functionals yield scaling-invariant regularity control. The analysis proves inevitable splash singularity formation within explicitly bounded time, maintaining spatial smoothness near the singular point up to the intersection time.

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arXiv2025

Classification of finite-time blow-up mechanisms for the incompressible free-boundary Euler equations with surface tension

We establish a blow-up criterion for strong solutions of the three-dimensional incompressible Euler equations with surface tension in a bounded domain with a closed moving free boundary. The criterion is formulated at the $H^3\times H^4$ regularity level of the Shatah--Zeng local well-posedness theory and imposes \textit{no} assumptions of symmetry, periodicity, graph structure, or simple connectedness. If the maximal existence time $T<\infty$, then at least one of the following four mechanisms must occur: (i) first self-intersection of the free boundary; (ii) loss of mean curvature regularity in $H^{\frac{3}{2}}$, or loss of boundary regularity in $H^{2+\varepsilon}$ for any sufficiently small fixed $\varepsilon>0$; (iii) loss of $H^{\frac{5}{2}}$ regularity of the normal boundary velocity; or (iv) $L^1_tL^\infty$ blow-up of the interior velocity gradient. For simply connected domains, the interior alternative admits a refinement involving only the $L^1_tL^\infty$-norm of the vorticity, and this refinement recovers exactly the classical Beale--Kato--Majda criterion in the fixed-boundary case. For irrotational flows in the simply connected free-boundary setting, the criterion reduces to the three boundary mechanisms.

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arXiv2024

Exponential stability of a free boundary problem with spherical symmetry for a gas bubble immersed in a bounded incompressible liquid

This paper 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.

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arXiv2024

Global well-posedness of the free boundary problem for incompressible viscous resistive MHD in critical Besov spaces

This paper aims to establish the global well-posedness of the free boundary problem for the incompressible viscous resistive magnetohydrodynamic (MHD) equations. Under the framework of Lagrangian coordinates, a unique global solution exists in the half-space provided that the norm of the initial data in the critical homogeneous Besov space $\dot{B}_{p, 1}^{-1+N/p}(\mathbb{R}_{+}^N)$ is sufficiently small, where $p \in [N, 2N-1)$. Building upon prior work such as (Danchin and Mucha, J. Funct. Anal. 256 (2009) 881--927) and (Ogawa and Shimizu, J. Differ. Equations 274 (2021) 613--651) in the half-space setting, we establish maximal $L^{1}$-regularity for both the Stokes equations without surface stress and the linearized equations of the magnetic field with zero boundary condition. The existence and uniqueness of solutions to the nonlinear problems are proven using the Banach contraction mapping principle.

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arXiv2023

A priori estimates and a blow-up criterion for the incompressible ideal MHD equations with surface tension and a closed free surface

We establish the a priori estimates and prove a blow-up criterion for the three-dimensional free boundary incompressible ideal magnetohydrodynamics equations. The fluid occupies a bounded region with a free boundary that is a closed surface, without assumptions of simple connectedness or periodicity of the region (thus, Fourier transforms cannot be applied), nor the graph assumption for the free boundary. The fluid is under the influence of surface tension, and flattening the boundaries using local coordinates is insufficient to resolve this problem. This is because local coordinates fail to preserve curvature, as the mean curvature of a flat boundary degenerates to zero. To address these challenges and circumvent the intricate issue of spatial regularity in Lagrangian coordinates, we utilize reference surfaces to represent the free boundary and develop new energy functionals that both preserve the material derivative and incorporate spatial-temporal scaling $\partial_t \sim \nabla^{\frac{3}{2}}$ in Eulerian coordinates. This method enables us to establish both low-order and high-order regularity estimates without any loss of regularity. More importantly, we prove a blow-up criterion and provide a complete classification of blow-ups, including the self-intersection of the free boundary (which the graph assumption cannot handle), the breakdown of the mean curvature, and the blow-up of the normal velocity (which Lagrangian coordinates fail to capture). To the best of our knowledge, this is the first result addressing the a priori estimates and the blow-up criterion for free boundary problems with surface tension in general regions.

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arXiv2023

Splash singularity for the free boundary incompressible viscous MHD

In this paper, we prove the existence of smooth initial data for the two-dimensional 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. The existence of the splash singularities is guaranteed by a local existence theorem, in which we need suitable spaces for the modified magnetic field and modification of the velocity and the pressure such that the modified initial velocity is zero, and a stability result which allows us to construct a class of initial velocities and domains for an arbitrary initial magnetic field. 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.

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arXiv2021

On the motion of free interface in two-dimensional ideal incompressible MHD

For the free boundary problem of the plasma-vacuum interface to ideal incompressible magnetohydrodynamics (MHD) in two-dimensional space, the a priori estimates of solutions are proved in Sobolev norms by adopting a geometrical point of view. In the vacuum region, the magnetic field is described by the div-curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic field is tangent to the boundary. We prove that the $L^2$ norms of any order covariant derivatives of the magnetic field in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall.

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arXiv2019

Well-posedness for the Linearized Free Boundary Problem of Incompressible Ideal Magnetohydrodynamics Equations

We study the well-posedness theory for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in the Lagrangian coordinates, and substitute the magnetic field into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, we linearize this equation with respect to the position vector field whose time derivative is the velocity, and obtain the local-in-time well-posedness of the solution by using energy estimates of the tangential derivatives and the curl with the help of Lie derivatives and the smooth-out approximation.

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arXiv2016

On the motion of free interface in ideal incompressible MHD

For the free boundary problem of the plasma-vacuum interface to three-dimensional ideal incompressible magnetohydrodynamics (MHD), the a priori estimates of smooth solutions are proved in Sobolev norms by adopting a geometrical point of view and some quantities such as the second fundamental form and the velocity of the free interface are estimated. In the vacuum region, the magnetic fields are described by the div-curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic fields are tangent to the interface, but we do not need any restrictions on the size of the magnetic fields on the free interface. We introduce the "virtual particle" endowed with a virtual velocity field in vacuum to reformulate the problem to a fixed boundary problem under the Lagrangian coordinates. The $L^2$-norms of any order covariant derivatives of the magnetic fields both in vacuum and on the boundaries are bounded in terms of initial data and the second fundamental forms of the free interface and the rigid wall. The estimates of the curl of the electric fields in vacuum are also obtained, which are also indispensable in elliptic estimates.

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arXiv2013

Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities

In this paper, we establish a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, the $γ$-gas law equation of state for $γ=2$ and the general initial density $\ri \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which play a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.

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arXiv2013

A priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows

In the present paper, we prove the a priori estimates of Sobolev norms for a free boundary problem of the incompressible inviscid MHD equations in all physical spatial dimensions $n=2$ and 3 by adopting a geometrical point of view used in Christodoulou-Lindblad CPAM 2000, and estimating quantities such as the second fundamental form and the velocity of the free surface. We identify the well-posedness condition that the outer normal derivative of the total pressure including the fluid and magnetic pressures is negative on the free boundary, which is similar to the physical condition (Taylor sign condition) for the incompressible Euler equations of fluids.

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arXiv2012

Well-posedness for a multi-dimensional viscous liquid-gas two-phase flow model

The Cauchy problem of a multi-dimensional ($d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is concerned in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local in time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.

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arXiv2009

Cauchy problem for viscous rotating shallow water equations

We consider the Cacuhy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modelling of motions for shallow water with free surface in a rotating sub-domain. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without coriolis forces, the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments.

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arXiv2008

Well-posedness for one-dimensional derivative nonlinear Schrödinger equations

In this paper, we investigate the one-dimensional derivative nonlinear Schrödinger equations of the form $iu_t-u_{xx}+iλ\abs{u}^k u_x=0$ with non-zero $λ\in \Real$ and any real number $k\gs 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.

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