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Ruilin Hu

Publications and source records attributed to Ruilin Hu.

10 recordsLinked to original sources
arXiv2026

Well-posedness for the mean curvature flow on the half-space and on bounded domains

We study graphical mean curvature flow in arbitrary codimension over the half-space and over smooth bounded domains, subject to homogeneous Dirichlet boundary conditions. At the scaling-critical Lipschitz regularity, we prove local well-posedness for initial graphs that can be approximated in $W^{1,\infty}$ by smooth profiles compatible with the boundary condition. Within this class, if the initial Lipschitz seminorm is sufficiently small, the corresponding solution is global; on a bounded domain, it also converges exponentially to the flat graph. Positive-time regularization is quantified by time-weighted H\"older estimates whose weighted quantities remain bounded as $t\downarrow0$. The main analytic ingredient is a boundary Schauder theory for variable-coefficient parabolic systems. On the half-space, it combines coefficient freezing with parity extensions and boundary identities intrinsic to the graphical system. On curved domains, localization and boundary flattening lead to anisotropic estimates, from which normal derivatives are recovered recursively.

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arXiv2026

Solutions of the 3D inhomogeneous incompressible Navier-Stokes system with initial velocity in $VMO^{-1}$

In this paper, we establish local existence of strong solutions for the three-dimensional inhomogeneous incompressible Navier-Stokes equations with initial data $(\rho_0,u_0)$ lying in $C^1 \times (L^2 \cap VMO^{-1})$, where $\rho_0$ has a positive lower bound. Furthermore, if $\rho_0 \in C^2$ and $||\rho_0-1||_{L^\infty}+||u_0||_{BMO^{-1}}$ is sufficiently small, we prove global existence of the solution. To achieve this, we employ an estimate for the transport equation to obtain regularity for the density and apply a new freezing-coefficient method for the momentum equation.

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arXiv2026

Schauder-type Estimates and Log-Critical Well-posedness for the Two-Phase Muskat Problem with Surface Tension

We prove short-time well-posedness for the Muskat problem with surface tension in the full two-phase setting, allowing different viscosities, arbitrary density contrast, and rigid boundaries. In particular, no Rayleigh--Taylor sign condition on the density contrast is imposed. The interface is assumed to be a graph, uniformly separated from the fixed boundaries, and the initial data may be large in the log-critical class $\dot C^{1,\log^\varkappa}\cap H^1$, with $\varkappa>1$. Thus the result reaches the natural Lipschitz threshold up to a logarithmic correction. The main difficulty is that, in the presence of viscosity jump and boundaries, the interface equation is not given by a closed explicit contour dynamics law. Instead, the normal velocity is recovered through an elliptic transmission problem in moving domains, and the resulting evolution is a genuinely nonlocal quasilinear equation. We derive sharp Schauder-type estimates, adapted to the log-critical scale, for the transmission operators generated by the bulk Darcy flow. These estimates identify the third-order parabolic mechanism produced by surface tension and control the nonlinear coupling between the interface geometry and the elliptic transmission structure. The proof builds on the Schauder framework developed in Part~I of this series \cite{CHN1}, but requires a new analysis of the Muskat transmission problem in moving domains. Combining this elliptic theory with the contour formulation and time-weighted H\"older estimates, we obtain existence, uniqueness, smoothing, and stability for large interfaces in arbitrary dimension.

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arXiv2026

Schauder-type Estimates and Well-posedness for Nonlocal Quasilinear Evolution Equations in Fluid Dynamics

We establish Schauder-type estimates for linear parabolic systems driven by variable-coefficient nonlocal pseudo-differential operators of order $s>0$. These estimates are formulated in critical time-weighted H\"older/Besov-type spaces and are tailored to quasilinear equations at scaling-critical regularity. A key ingredient is a kernel-adapted freezing-coefficient method. After freezing the coefficients at a reference point, we derive explicit representation formulas through the corresponding fundamental kernels and then evaluate the resulting bounds at the physical point. This avoids treating the coefficient variation as a separate lower-order perturbation and yields robust control of the residual terms within the leading-order dynamics. As an application, we obtain a general well-posedness framework for a class of nonlocal quasilinear parabolic equations in critical spaces. In particular, we prove critical local and, in suitable regimes, global well-posedness for the Muskat equation with surface tension and for the two- and three-dimensional Peskin problems with nonlinear elastic tension. These results provide a unified critical framework for distinct nonlocal evolution equations arising in fluid dynamics and related areas.

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arXiv2025

Well-posedness of the Fractional Fokker-Planck Equation

In this paper, we employ a Schauder-type estimate method, as developed in \cite{CHN}, to establish critical well-posedness result for the Fractional Fokker-Planck Equation. This equation serves as a fundamental model in kinetic theory and can be regarded as a semi-linear analogue of the non-cutoff Boltzmann equation. We demonstrate that the techniques introduced in this study are not only effective for the FFPE but also hold promise for broader applications, particularly in addressing the non-cutoff Boltzmann equation and the Landau equation. Our results contribute to a deeper understanding of the analytical framework required for these complex kinetic models.

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arXiv2024

Quantitative bounds for bounded solutions to the Navier-Stokes equations in endpoint critical Besov spaces

Building on Tao's quantitative regularity theory and triple-logarithmic blow-up estimate in $L^3$ in \cite{Tao_20}, we consider classical solutions $(u,P)$ of the three-dimensional incompressible Navier--Stokes equations on $[0,T]\times\mathbb{R}^3$. For $3<p<\infty$, under simultaneous uniform control of the two scaling-critical quantities $\|u\|_{L_T^\infty(\dot B_{p,\infty}^{-1+\frac{3}{p}})}$ and $\||D|^{-1+\frac{3}{p}}u\|_{L_T^\infty(L^p)}$, we obtain explicit quantitative estimates for all spatial derivatives of $u$. As a consequence, we derive a mixed blow-up criterion coupling a double exponential of the critical Besov norm with the $L^p$ norm of $|D|^{-1+\frac{3}{p}}u$, which forces quantified growth of at least one of these two critical quantities near any finite blow-up time. The low regularity and lack of dyadic summability in the endpoint Besov space are handled through a finite iterative decomposition that successively improves spatial integrability and produces an energy-class remainder, together with refined nonlinear energy estimates. The nonlocal signed quantity $|D|^{-1+\frac{3}{p}}u$ is treated by localized mean-zero vector tests and almost orthogonality across geometrically separated concentration scales.

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arXiv2024

The Muskat problem with a large slope

In this paper, we establish local well-posedness results for the Muskat equation in any dimension using modulus of continuity techniques. By introducing a novel quantity \(\beta_\sigma(f_0')\) which encapsulates local monotonicity and slope, we identify a new class of initial data within \(W^{1,\infty}(\mathbb{R}^d)\). This includes scenarios where the product of the maximal and minimal slopes is large, thereby guaranteeing the local existence of a classical solution.

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arXiv2024

Well-posedness for local and nonlocal quasilinear evolution equations in fluids and geometry

We establish a Schauder-type estimate for general local and non-local linear parabolic system $$\partial_tu+\mathbf{L}_su=\Lambda^\gamma f+g$$ in $(0,\infty)\times\mathbb{R}^d$ where $\Lambda=(-\Delta)^{\frac{1}{2}}$, $0<\gamma\leq s$, $\mathbf{L}_s$ is the Pesudo-differential operator defined by \begin{equation} \mathbf{L}_su(t,x)=(2\pi)^{-\frac{d}{2}}\int_{\mathbb{R}^d}\mathsf{A}(t,x,\xi)\hat u(t,\xi)e^{ix\cdot\xi}d\xi,\quad\quad \mathsf{A}(t,x,\xi)\sim |\xi|^s. \end{equation} To prove this, we develop a new freezing coefficient method for kernel, where we freeze the coefficient at $x_0$, then derive a representation formula of the solution, and finally we take $x_0=x$ when estimating the solution. By applying our Schauder-type estimate to suitably chosen differential operators $\mathcal{L}_s$, we obtain critical well-posedness results of various local and non-local nonlinear evolution equations in geometry and fluids, including hypoviscous Navier--Stokes equations, the surface quasi-geostrophic equation, mean curvature equations, Willmore flow, surface diffusion flow, Peskin equations, thin-film equations and Muskat equations.

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arXiv2022

Global well-posedness of the 1d compressible Navier-Stokes system with rough data

In this paper, we study the global well-posedness problem for the 1d compressible Navier-Stokers system (cNSE) in gas dynamics with rough initial data. Frist, Liu- Yu (2022) established the global well-posedness theory for the 1d isentropic cNSE with initial velocity data in BV space. Then, it was extended to the 1d cNSE for the polytropic ideal gas with initial velocity and temperature data in BV space by Wang-Yu-Zhang (2022). We improve the global well-posedness result of Liu-Yu with initial velocity data in $W^{2γ,1}$ space; and of Wang-Yu-Zhang with initial velocity data in $ L^2\cap W^{2γ,1}$ space and initial data of temperature in $\dot W^{-\frac{2}{3},\frac{6}{5}}\cap \dot W^{2γ-1,1}$ for any $γ>0$ \textit{arbitrary small}. Our essential ideas are based on establishing various "end-point" smoothing estimates for the 1d parabolic equation.

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