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Yunrui Zheng

Publications and source records attributed to Yunrui Zheng.

9 recordsLinked to original sources
arXiv2026

Global regularity and general-coefficient singular limits for energy-critical complex Ginzburg--Landau equations

We study energy-critical complex Ginzburg--Landau equations with a linear damping term $Ru$, $R\geq 0$. For the undamped aligned equation in dimensions $d=3,4$, we treat the focusing and defocusing cases in a unified way and prove persistence of $H^1\cap C_0$ regularity and smoothness for positive times. In particular, this resolves the energy-critical cases of Cazenave's open problem. In dimensions $3\le d\le6$, we develop a coefficient-uniform critical stability framework for the zero-dispersion and inviscid limits. It applies to independent normalized complex coefficient paths in both the focusing and defocusing cases. From limiting data in the natural energy space $H^1$, we obtain convergence on every compact subinterval of the maximal lifespan of the limiting solution. Higher regularity is required only for explicit linear coefficient-error estimates, and the limits do not use global well-posedness, scattering, or global spacetime bounds for the limiting solution. At the inviscid limit, we establish coefficient-uniform homogeneous and retarded Strichartz estimates; a key technical ingredient is the retarded double-endpoint estimate required by the critical forcing space.

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arXiv2026

Global Well-Posedness of Contact Lines: 2D Navier-Stokes Flow

Based on the global a priori estimates in [Guo-Tice, J. Eur. Math. Soc. (2024)], we establish the well-posedness of a viscous fluid model satisfying the dynamic law for the contact line \begin{equation*} \mathscr{W}(\p_tζ(\pm\ell,t))=[\![γ]\!]\mpσ\frac{\p_1ζ}{(1+|\p_1ζ|^2)^{1/2}}(\pm\ell,t) \end{equation*} in 2D domain, where $ζ(x_1,t)$ is a free surface with two contact points $ζ(\pm\ell,t)$, $[\![γ]\!]$ and $σ$ are constants characterizing the solid-fluid-gas free energy, and the increasing $\mathscr{W}$ is the contact point velocity response function. Motivated by the energy-dissipation structure, our construction relies on the construction of a pressureless weak solution for the coupled velocity and free interface for the linearized problems, via a Galerkin approximation with a time-dependent basis and an artificial regularization for the capillary operator.

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arXiv2026

Global weak solution of 3-D focusing energy-critical nonlinear Schrödinger equation

In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schrödinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some class of initial data. The main ingredient of our new approach is to use solutions of an energy-critical Ginzburg-Landau equation as approximations for the corresponding nonlinear Schördinger equation. In our proofs, we first show the dichotomy of global well-posedness versus finite time blow-up of energy-critical Ginzburg-Landau equation in $\dot{H}^1( \mathbb{R}^d)$ for $d = 3,4 $ when the energy is less than the energy of the stationary solution $W$. We follow the strategy of C. E. Kenig and F. Merle [25,26], using a concentration-compactness/rigidity argument to reduce the global well-posedness to the exclusion of a critical element. The critical element is ruled out by dissipation of the Ginzburg-Landau equation, including local smoothness, backwards uniqueness and unique continuation. The existence of global weak solution of the three dimensional focusing energy-critical nonlinear Schrödinger equation in the non-radial case then follows from the global well-posedness of the energy-critical Ginzburg-Landau equation via a limitation argument. We also adapt the arguments of M. Struwe [37,38] to prove the weak-strong uniqueness when the $\dot{H}^1$-norm of the initial data is bounded by a constant depending on the stationary solution $W$.

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arXiv2025

Dynamics of contact points in 2D Boussinesq flow

We consider the evolution of contact lines for thermal convection of viscous fluids in a 2D open-top vessel. The domain is bounded above by a free moving boundary and otherwise by the solid wall of a vessel. The dynamics of the fluid are governed by the incompressible Boussinesq approximation under the influence of gravity, and the interface between fluid and air is under the effect of capillary forces. Motivated by energy-dissipation structure in [Guo-Tice, J. Eur. Math. Soc, 2024], we develop global well posedness theory in the framework of nonlinear energy methods for the initial data sufficiently close to equilibrium. Moreover, the solutions decay to equilibrium at an exponential rate. Our methods are mainly based on the construction of solutions to convected heat equation and a priori estimates of a geometric formulation of the Boussinesq equations.

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arXiv2024

Global well-posedness of the energy-critical complex Ginzburg-Landau equation in exterior domains

In this article, we consider an energy-critical complex Ginzburg-Landau equation in the exterior of a smooth compact strictly convex obstacle. We prove the global well-posedness of energy-critical complex Ginzburg-Landau equation in an exterior domain by the concentration-compactness/rigidity theorem method. As corollaries of our main result, we establish both the existence of global weak solutions to energy-critical defocusing nonlinear Schrödinger equations and the global well-posedness theory for energy-critical semi-linear heat equation in exterior domains.

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arXiv2024

Stability of Navier-Stokes equations with a free surface

We consider the viscous incompressible fluids in a three-dimensional horizontally periodic domain bounded below by a fixed smooth boundary and above by a free moving surface. The fluid dynamics are governed by the Navier-Stokes equations with the effect of gravity and surface tension on the free surface. We develop a global well-posedness theory by a nonlinear energy method in low regular Sobolev spaces with several techniques, including: the horizontal energy-dissipation estimates, a new tripled bootstrap argument inspired by Guo and Tice [Arch. Ration. Mech. Anal.(2018)]. Moreover, the solution decays asymptotically to the equilibrium in an exponential rate.

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arXiv2016

Local well-posedness of the contact line problem in 2-D Stokes flow

We consider the evolution of contact lines for viscous fluids in a two-dimensional open-top vessel. The domain is bounded above by a free moving boundary and otherwise by the solid wall of a vessel. The dynamics of the fluid are governed by the incompressible Stokes equations under the influence of gravity, and the interface between fluid and air is under the effect of capillary forces. Here we develop a local well-posedness theory of the problem in the framework of nonlinear energy methods. We utilize several techniques, including: energy estimates of a geometric formulation of the Stokes equations, a Galerkin method with a time-dependent basis for an $ε$--perturbed linear Stokes problem in moving domains, the contraction mapping principle for the $ε$--perturbed nonlinear full contact line problem, and a continuity argument for uniform energy estimates.

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arXiv2015

Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary

In this paper, we prove the local well-posedness of the free boundary problem for the incompressible Euler equations in low regularity Sobolev spaces, in which the velocity is a Lipschtiz function and the free surface belongs to $C^{\f32+\varepsilon}$. Moreover, we also present a Beale-Kato-Majda type break-down criterion of smooth solution in terms of the mean curvature of the free surface, the gradient of the velocity and Taylor sign condition.

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