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Dongyi Wei

Publications and source records attributed to Dongyi Wei.

At least 19 recordsLinked to original sources
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

Critical mass threshold for the 2D Patlak-Keller-Segel-Navier-Stokes system

In this paper, we investigate critical mass threshold for the Patlak-Keller-Segel-Navier-Stokes system on the two-dimensional whole space and obtain global existence of strong solutions if the initial mass is less than or equal to $8\pi$, regardless of the initial norm of the velocity. One new observation is that the local mass of the density function rearrangement satisfies a good inequality that is independent of velocity; and then an improved maximum principle is applied by choosing a nice auxiliary function.

math.AP↗
arXiv2026

Self-similar blow-up solutions of $d$-dimensional incompressible Euler equations with $C^{1,\left(1-2/d\right)-}$ velocity

We investigate self-similar blow-up solutions to the $d$-dimensional axisymmetric incompressible Euler equations without swirl for $d\ge 3$. For any $\alpha\in(0, \alpha_d)$ with $\alpha_d=1-2/d$, we construct a self-similar blow-up solution whose initial velocity field satisfies $u_0\in C^{1,\alpha}_{\rm loc}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})$. Our construction relies on a fixed-point argument formulated for the self-similar profile equations, which form a coupled elliptic-transport system. Specifically, the transport equation recovers the vorticity profile from given data along characteristic curves, while the elliptic equation reconstructs the velocity field via Newtonian potentials defined in an auxiliary $(d+4)$-dimensional space. The main challenge consists in choosing appropriate function spaces that remain invariant under such nonlinear compositions and that simultaneously capture the exact singular behavior near the origin and the symmetry axis. Furthermore, we establish a finite-codimensional stability result for the self-similar profiles obtained above. As a consequence, after suitable truncation and correction of finitely many unstable modes, we obtain finite-energy blow-up solutions with initial velocity in $C^{1,\alpha}(\mathbb R^d)\cap C^\infty(\mathbb R^d\setminus\{0\})\cap L^2(\mathbb R^d)$ and compactly supported initial vorticity. These solutions are asymptotically self-similar near the blow-up time.

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arXiv2026

Flexible exponent of geometric 3-manifolds and Legendrian maps of Seifert spaces

A classical question in quantitative topology is to bound the mapping degree $\operatorname{deg}(f)$ in terms of its Lipchitz constant $\operatorname{Lip}(f)$. For a closed, oriented manifold $M$, the flexible exponent $\alpha(M)$ is the infimum of $\alpha\geq 0$ such that $|\operatorname{deg} f|\leq C(\operatorname{Lip} f)^\alpha$ holds for all differentiable map $f:M\to M$. The flexible exponent measures how effectively a manifold can wrap itself through self-maps. For geometric 3-manifolds $M$ in the sense of Thurston, we give the complete result for $\alpha(M)$: \[ \alpha(M)= \begin{cases} 3 & M \text{ modeled on } \mathbb S^3,\mathbb E^3,\mathbb S^2\times\mathbb E^1,\\ \frac83 & M \text{ modeled on Nil},\\ 2 & M \text{ modeled on Sol},\\ 1 & M \text{ modeled on }\mathbb H^2\times\mathbb E^1,\\ 0 & M \text{ modeled on } \mathbb H^3,\widetilde{\rm SL_2}. \end{cases} \] To prove $\alpha(M)=8/3$ for Nil 3-manifold $M$, we construct the so-called Legendrian map: a smooth self-map $f: M\to M$ such that $f$ is homotopic to the identity and $f$ maps all $S^1$-fibers into the orthogonal contact plane field simultaneously. Moreover, we prove that any Legendrian map must not be a diffeomorphism.

math.GT↗
arXiv2025

Transition threshold for the Navier-Stokes-Coriolis system at high Reynolds numbers

The transition mechanism from laminar flow to turbulent flow is a central problem in hydrodynamic stability theory. To shed light on this transition mechanism, Trefethen et al.({\it \small Science 1993}) proposed the transition threshold problem, aiming to quantify the magnitude of perturbations required to trigger instability and determine their scaling with the Reynolds number. In this paper, we investigate the transition threshold of Couette flow for the three-dimensional incompressible Navier-Stokes-Coriolis system in the high Reynolds number regime ($\mathrm{Re}\gg 1$). By exploiting the combined effects of rotation (dispersion) and mixing mechanisms, we derive an improved stability threshold scaling in $\mathrm{Re}$. Precisely, we show that if the initial perturbation satisfies $$\|v_{in}-(y, 0, 0)\|_{\tilde{H}(\mathbb T \times \mathbb D)}\leq \epsilon_0 \,\mathrm{Re}^{-\alpha},$$ with any $\alpha>\frac 23$ and $\tilde{H}=H^6 \cap W^{3,1}$ for $\mathbb D=\mathbb{R}^2$, and with any $\alpha \geq\frac 56$ and $\tilde{H}=H^6$ for $\mathbb D=\mathbb{R}\times\mathbb{T}$, the corresponding solution of the Navier-Stokes-Coriolis system exists globally in time and remains asymptotically close to the Couette flow. The main analytical challenge arises from the anisotropic nature of the estimates for the zero modes and from the interactions between zero and non-zero modes, which we address using an anisotropic Sobolev space directly tailored to the zero modes. Additionally, we introduce a new dispersive structure for the zero modes and derive suitable Strichartz-type estimates. These tools enable us to exploit both the nonlinear structure and the improved dispersive behavior of certain good components of the zero modes, which play a crucial role in achieving the improved stability threshold.

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arXiv2025

The Immersed Boundary Problem in 2-D: the Navier-Stokes Case

We study the immersed boundary problem in 2-D. It models a 1-D elastic closed string immersed and moving in a fluid that fills the entire plane, where the fluid motion is governed by the 2-D incompressible Navier-Stokes equation with a positive Reynolds number subject to a singular forcing exerted by the string. We introduce the notion of mild solutions to this system, and prove its existence, uniqueness, and optimal regularity estimates when the initial string configuration is $C^1$ and satisfies the well-stretched condition and when the initial flow field $u_0$ lies in $L^p(\mathbb{R}^2)$ with $p\in (2,\infty)$. A blow-up criterion is also established. When the Reynolds number is sent to zero, we show convergence in short time of the solution to that of the Stokes case of 2-D immersed boundary problem, with the optimal error estimates derived. We prove the energy law of the system when $u_0$ additionally belongs to $L^2(\mathbb{R}^2)$. Lastly, we show that the solution is global when the initial data is sufficiently close to an equilibrium state.

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arXiv2025

Asymptotic stability of the Kolmogorov flow at high Reynolds numbers

In this paper we prove the asymptotic stability of the Kolmogorov flow on a non-square torus for perturbations $\omega_0$ satisfying $\|\omega_0\|_{H^3}\ll\nu^{1/3}$, where $0<\nu\ll1$ is the viscosity. Kolmogorov flows are important metastable states to the two dimensional incompressible Navier Stokes equations in the high Reynolds number regime. Our result shows that the perturbed solution will rapidly converge to a shear flow close to the Kolmogorov flow, before settling down to the Kolmogorov flow and slowly decaying to $0$ as $t\to\infty$. In fact, our analysis reveals several interesting time scales and rich dynamical behavior of the perturbation in the transition period $0<t\leq 1/\nu$. The threshold $\nu^{1/3}$, which is the same as that for the Couette flow, is quite surprising since one of the key stability mechanisms, enhanced dissipation, becomes considerably weaker in the case of Kolmogorov flows due to the presence of critical points. To overcome this essential new difficulty, we establish sharp vorticity depletion estimates near the critical points to obtain improved decay rates for the vorticity and velocity fields that are comparable with those for Couette flows, at least for our purposes. We then combine these estimates (enhanced dissipation, inviscid damping and vorticity depletion) with a quasilinear approximation scheme and a multiple-timescale analysis naturally adapted to the dynamics of the perturbation, to obtain the $\nu^{1/3}$ threshold for dynamic stability of Kolmogorov flows. The threshold is expected to be sharp when the perturbation is considered in Sobolev spaces. This appears to be the first result that applies vorticity depletion estimates to improve thresholds for nonlinear asymptotic stability in incompressible fluid equations.

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arXiv2025

Self-similar algebraic spiral vortex sheets of 2-D incompressible Euler equations

This paper provides the first rigorous construction of the self-similar algebraic spiral vortex sheet solutions to the 2-D incompressible Euler equations. These solutions are believed to represent the typical roll-up pattern of vortex sheets after the formation of curvature singularities. The most challenging part of this paper is to handle the Cauchy integral for the algebraic spiral curve, which falls outside the classical theory of singular integral operators.

math.AP↗
arXiv2025

Blow-up of the 3-D compressible Navier-Stokes equations for monatomic gases

In this paper, we prove the blow-up of the $3$-D isentropic compressible Navier-Stokes equations for the adiabatic exponent $\gamma=5/3$, which corresponds to the law of monatomic gases. This is the degenerate case in the sense of [Merle, Rapha\"el, Rodnianski and Szeftel, Ann. of Math. (2), 196 (2022), 567-778; Ann. of Math. (2), 196 (2022), 779-889]. Motivated by these breakthrough works, we first establish the existence of a sequence of smooth, self-similar imploding solutions to the compressible Euler equations for $\gamma=5/3$. Subsequently, we utilize these self-similar profiles to construct smooth, asymptotically self-similar blow-up solutions to the compressible Navier-Stokes equations for monatomic gases.

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arXiv2024

Global well-posedness and self-similar solution of the inhomogeneous Navier-Stokes system

In this paper, we study the global well-posedness of the 3-D inhomogeneous incompressible Navier-Stokes system (INS in short) with initial density $\rho_0$ being discontinuous and initial velocity $u_0$ belonging to some critical space. Firstly, if $\rho_0u_0$ is sufficiently small in the space $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}(\mathbb{R}^3)$ and $\rho_0$ is close enough to a positive constant in $L^\infty$, we establish the global existence of strong solution to (INS) for $3<p<\infty$ and provide the uniqueness of the solution for $3<p<6$. This result corresponds to Cannone-Meyer-Planchon solution of the classical Navier-Stokes system. Furthermore, with the additional assumption that $u_0\in L^2(\mathbb{R}^3)$, we prove the weak-strong uniqueness between Cannone-Meyer-Planchon solution and Lions weak solution of (INS). Finally, we prove the global well-posedness of (INS) with $u_0\in \dot{B}^{\frac{1}{2}}_{2,\infty}(\mathbb{R}^3)$ being small and only an upper bound on the density. This gives the first existence result of the forward self-similar solution for (INS).

math.AP↗
arXiv2024

A new proof of nonlinear Landau damping for the 3D Vlasov-Poisson system near Poisson equilibrium

This paper investigates nonlinear Landau damping in the 3D Vlasov-Poisson (VP) system. We study the asymptotic stability of the Poisson equilibrium $\mu(v)=\frac{1}{\pi^2(1+|v|^2)^2}$ under small perturbations. Building on the foundational work of Ionescu, Pausader, Wang, and Widmayer \cite{AIonescu2022}, we provide a streamlined proof of nonlinear Landau damping for the 3D unscreened VP system. Our analysis leverages sharp decay estimates, novel decomposition techniques to demonstrate the stabilization of the particle distribution and the decay of electric field. These results reveal the free transport-like behavior for the perturbed density $\rho(t,x)$, and enhance the understanding of Landau damping in an unconfined setting near stable equilibria.

math.AP↗
arXiv2024

Self-similar finite-time blowups with smooth profiles of the generalized Constantin-Lax-Majda model

We show that the $a$-parameterized family of the generalized Constantin-Lax-Majda model, also known as the Okamoto-Sakajo-Wunsch model, admits exact self-similar finite-time blowup solutions with interiorly smooth profiles for all $a\leq 1$. Depending on the value of $a$, these self-similar profiles are either smooth on the whole real line or compactly supported and smooth in the interior of their closed supports. The existence of these profiles is proved in a consistent way by considering the fixed-point problem of an $a$-dependent nonlinear map, based on which detailed characterizations of their regularity, monotonicity, and far-field decay rates are established. Our work unifies existing results for some discrete values of $a$ and also explains previous numerical observations for a wide range of $a$.

math.AP↗
arXiv2024

Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities

The dynamics of probability density functions have been extensively studied in computational science and engineering to understand physical phenomena and facilitate algorithmic design. Of particular interest are dynamics formulated as gradient flows of energy functionals under the Wasserstein metric. The development of functional inequalities, such as the log-Sobolev inequality, plays a pivotal role in analyzing the convergence of these dynamics. This paper aims to extend the success of functional inequality techniques to dynamics that are gradient flows under the Fisher-Rao metric, with various $f$-divergences serving as energy functionals. Such dynamics take the form of nonlocal differential equations, for which existing analyses critically rely on explicit solution formulas in special cases. We provide a comprehensive study of functional inequalities and the relevant geodesic convexity for Fisher-Rao gradient flows under minimal assumptions. A notable feature of our functional inequalities is their independence from the log-concavity or log-Sobolev constants of the target distribution. Consequently, the convergence rate of the dynamics (assuming well-posedness) remains uniform across general target distributions.

math.AP↗
arXiv2024

Exponential mixing for random nonlinear wave equations: weak dissipation and localized control

We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics, i.e., asymptotic compactness in dynamical systems, global stability of evolution equations, and localized control problems. As an initial application, we exploit the exponential mixing of random nonlinear wave equations with degenerate damping, critical nonlinearity, and physically localized noise. The essential challenge lies in the fact that the weak dissipation and randomness interact in the evolution.

math.AP↗
arXiv2024

Global well-posedness of inhomogeneous Navier-Stokes equations with bounded density

In this paper, we solve Lions' open problem: {\it the uniqueness of weak solutions for the 2-D inhomogeneous Navier-Stokes equations (INS)}. We first prove the global existence of weak solutions to 2-D (INS) with bounded initial density and initial velocity in $L^2(\mathbb R^2)$. Moreover, if the initial density is bounded away from zero, then our weak solution equals to Lions' weak solution, which in particular implies the uniqueness of Lions' weak solution. We also extend a celebrated result by Fujita and Kato on the 3-D incompressible Navier-Stokes equations to 3-D (INS): {\it the global well-posedness of 3-D (INS) with bounded initial density and initial velocity being small in $\dot H^{1/2}(\mathbb R^3)$}. The proof of the uniqueness is based on a surprising finding that the estimate $t^{1/2}\nabla u\in L^2(0,T; L^\infty(\mathbb R^d))$ instead of $\nabla u\in L^1(0, T; L^\infty(\mathbb R^d))$ is enough to ensure the uniqueness of the solution.

math.AP↗
arXiv2024

On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations

In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumption that the initial density is only bounded and the initial velocity is in $H^1(\mathbb{R}^2)$. With suitable assumptions on the initial density, which includes the case of density patch and vacuum bubbles, we prove that Lions' s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions: {\it for the density patch data $ρ_0=1_{D}$ with a smooth bounded domain $D\subset\mathbb{R}^2$, the regularity of $D$ is preserved by the time evolution of Lions's weak solution.}

math.AP↗
arXiv2024

On blow-up for the supercritical defocusing nonlinear wave equation

In this paper, we consider the defocusing nonlinear wave equation $-\partial_t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb R\times \mathbb R^d$. Building on our companion work ({\it \small Self-similar imploding solutions of the relativistic Euler equations}), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$, there exists a smooth complex-valued solution that blows up in finite time.

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