arXiv subjects

Lizhe Wan

Publications and source records attributed to Lizhe Wan.

13 recordsLinked to original sources

arXiv2026

Self-similar solutions of the three-dimensional Muskat problem with surface tension

We construct a one-parameter family of small self-similar solutions to the three-dimensional one-phase Muskat problem with surface tension. The solutions have the form $\eta_\varepsilon(t,x) = t^{1/3}U_\varepsilon(t^{-1/3}x)$ and emanate from the conical initial data $\eta_\varepsilon(0,x)=\varepsilon|x|$. The profiles are perturbations of the linear capillary regularization of the cone, and we identify the leading quadratic correction. The proof combines a raywise inverse estimate for the linear similarity operator, a favorable high--high-to-low cancellation in the quadratic term, and finite-order tame estimates for the Dirichlet--Neumann operator on asymptotically conical graphs. These estimates yield the solutions by a contraction argument and show that the conical singularity is instantaneously rounded for positive time.

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arXiv2026

Nonlinear modulational instability of two-dimensional deep hydroelastic Stokes waves

In this paper, we study the nonlinear modulational instability of two-dimensional hydroelastic Stokes waves in infinite depth. We first justify a focusing cubic nonlinear Schr\"odinger (NLS) approximation result for 2D deep hydroelastic wave system in the spirit of Ifrim-Tataru [22]. Then we exploit the instability mechanism of the cubic NLS to prove that the Stokes waves are nonlinearly unstable under long-wave perturbations.

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arXiv2026

Exact controllability of two-dimensional hydroelastic waves

We prove the exact controllability of two-dimensional hydroelastic waves in the periodic setting. We show that if the initial data and the final data are small, for exterior pressure whose support is any non-empty open set $\omega$, the two-dimensional hydroelastic wave system is exactly controllable in arbitrary short time.

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arXiv2026

On the well-posedness of two-dimensional Muskat problem with an elastic interface

We investigate the two-dimensional Muskat problem with a nonlinear elastic interface, for both one-phase and two-phase scenarios. Following the framework developed by Nguyen [35,36], we demonstrate that the problem is locally well-posed in $H^s$ for $s\geq 2$ for arbitrary initial data. Furthermore, for the one-phase case and the stable two-phase case $(\rho^+ \leq \rho^-)$, we establish global well-posedness for small initial data in $H^s$ when $s> \frac{3}{2}$.

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arXiv2025

Low regularity well-posedness for two-dimensional hydroelastic waves

We investigate the low regularity local well-posedness of two-dimensional irrotational deep hydroelastic waves. Building on the approach of Ifrim-Tataru [29] and Ai-Ifrim-Tataru [5], in particular by constructing a cubic modified energy that incorporates a paradifferential weight chosen carefully, we prove that the hydroelastic waves are locally well-posed in $\mathcal{H}^s$ for $s>\frac{3}{4}$.

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arXiv2025

On the gravity-capillary water waves with point vortices

We consider the two-dimensional deep gravity-capillary water waves with point vortices. We first formulate the question in the holomorphic coordinates. Then, we derive an a priori energy estimate for water waves, and show that the water wave system has a unique solution for initial data in $\mathcal{H}^s\times \Omega_t^N$, $s>\frac{3}{2}$. Finally, we show that if there are only two vortices, and vortices, initial velocity, and initial surfaces are symmetric with respect to the vertical axis, then the solution of pure capillary water waves with two point vortices has an extended cubic lifespan.

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arXiv2024

On the capillary water waves with constant vorticity

This article is devoted to the study of local well-posedness for deep water waves with constant vorticity in two space dimensions on the real line. The water waves can be paralinearized and written as a quasilinear dispersive system of equations. By using the energy estimate and the Strichartz estimate, we show that for $s> \frac{5}{4}$, the gravity-capillary water wave system with constant vorticity is locally well-posed in $\mathcal{H}^{s}(\mathbb{R})$.

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arXiv2024

Low regularity well-posedness for two-dimensional deep water waves

The study of gravity-capillary water waves in two space dimensions has been an important question in mathematical fluid dynamics. By implementing the cubic modified energy method of Ifrim-Tataru in the context of gravity-capillary waves, we show that for $s> 1$, the two-dimensional gravity-capillary water wave system is locally well-posed in $\mathcal{H}^{s}$.

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arXiv2024

Two-dimensional solitary water waves with constant vorticity, Part II: the deep capillary case

We consider the two-dimensional capillary water waves with nonzero constant vorticity in infinite depth. We first derive the Babenko equation that describes the profile of the solitary wave. When the velocity $c$ is close to a critical velocity and a sign condition involving the physical parameters is met, the Babenko equation can be reduced to the stationary focusing cubic nonlinear Schr\"odinger equation plus perturbative error. We show the existence of a critical value of a dimensionless physical parameter below which at least two families of velocities satisfy the focusing condition and above which only one does. This gives the existence of small-amplitude solitary wave solutions for the water wave system with constant vorticity.

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arXiv2023

Low regularity well-posedness for two-dimensional deep gravity water waves with constant vorticity

We consider the two dimensional gravity water waves with nonzero constant vorticity in infinite depth. We show that for $s\geq \frac{3}{4}$, the water waves system is locally well-posed in $\mathcal{H}^{s}$, which is the nonzero constant vorticity counterpart of the breakthrough work of Ai-Ifrim-Tataru in [4]. It is also a $\frac{1}{4}$ improvement in Sobolev regularity compared to the previous result of Ifrim-Tataru in [17].

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arXiv2023

Two dimensional solitary water waves with constant vorticity, Part I: the deep gravity case

We consider the two dimensional pure gravity water waves with nonzero constant vorticity in infinite depth, working in the holomorphic coordinates introduced by Hunter, Ifrim, and Tataru. We show that close to the critical velocity corresponding to zero frequency, a solitary wave exists. We use a fixed point argument to construct the solitary wave whose profile resembles a rescaled Benjamin-Ono soliton. The solitary wave is smooth and has an asymptotic expansion in terms of powers of the Benjamin-Ono soliton.

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arXiv2023

On the $L^2$ well-posedness and decay estimate of third order Benjamin-Ono equation

We consider the $L^2$ well-posedness of third order Benjamin-Ono equation. We show that by means of a normal form and a gauge transformation, the equation can be changed into an Airy-type equation. A second goal of this work is to establish that the solutions to the nonlinear third order Benjamin-Ono equation problem exhibit a dispersive decay estimate analogue to the corresponding linear associated problem. The key ingredient is the use of a nonlinear vector field method.

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arXiv2022

The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity

This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.

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