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Quoc-Hung Nguyen

Publications and source records attributed to Quoc-Hung Nguyen.

At least 37 records · Page 2Linked to original sources
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 $μ(v)=\frac{1}{π^2(1+|v|^2)^2}$ 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 $ρ(t,x)$, and enhance the understanding of Landau damping in an unconfined setting near stable equilibria.

math.AP↗
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 \(β_σ(f_0')\) 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.

math.AP↗
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 $$\partial_tu+\mathbf{L}_su=Λ^γf+g$$ in $(0,\infty)\times\mathbb{R}^d$ where $Λ=(-Δ)^{\frac{1}{2}}$, $0<γ\leq s$, $\mathbf{L}_s$ is the Pesudo-differential operator defined by \begin{equation} \mathbf{L}_su(t,x)=(2π)^{-\frac{d}{2}}\int_{\mathbb{R}^d}\mathsf{A}(t,x,ξ)\hat u(t,ξ)e^{ix\cdotξ}dξ,\quad\quad \mathsf{A}(t,x,ξ)\sim |ξ|^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.

math.AP↗
arXiv2023

Global regularity for critical SQG in bounded domains

We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in $\mathbb R^2$. This solves an open problem. We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.

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arXiv2023

Potential theory for nonlocal drift-diffusion equations

The purpose of this paper is to prove new fine regularity results for nonlocal drift-diffusion equations via pointwise potential estimates. Our analysis requires only minimal assumptions on the divergence free drift term, enabling us to include drifts of critical order belonging merely to BMO. In particular, our results allow to derive new estimates for the dissipative surface quasi-geostrophic equation.

math.AP↗
arXiv2023

An optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ and its consequences

For $n > k \geq 0$, $λ>0$, and $p, r>1$, we establish the following optimal Hardy-Littlewood-Sobolev inequality \[ \Big| \iint_{\mathbf R^n \times \mathbf R^{n-k}} \frac{f(x) g(y)}{ |x-y|^λ|y"|^β} dx dy \Big| \lesssim \| f \| _{L^p(\mathbf R^{n-k})} \| g\| _{L^r(\mathbf R^n)} \] with $y = (y', y") \in \mathbf R^{n-k} \times \mathbf R^k$ under the two necessary conditions \[ β< \left\{ \begin{aligned} & k - k/r & & \text{if } \; 0 < λ\leq n-k,\\ & n - λ- k/r & & \text{if } \; n-k < λ, \end{aligned} \right. \] and \[ \frac{n-k}n \frac 1p + \frac 1r + \frac { β+ λ} n = 2 -\frac kn. \] We call this the optimal Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$. The existence of an optimal pair for this new inequality is also studied. The motivation of working on the above inequality is to provide a unification of many known Hardy-Littewood-Sobolev inequalities including the classical Hardy-Littewood-Sobolev inequality when $k=β=0$, the Hardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1} \times \mathbf R_+^n$ when $k=1$ and $β= 0$, and the Hardy-Littewood-Sobolev inequality on the upper half space $\mathbf R^{n-1} \times \mathbf R_+^n$ with extended kernel when $k=1$ and $β\ne 0$. We show that the above condition for $β$ is sharp. In the unweighted case, namely $β=0$, our finding immediately leads to the sharp Hardy-Littlewood-Sobolev inequality on $\mathbf R^{n-k} \times \mathbf R^n$ with the optimal range $$0<λ<n-k/r,$$ which has not been observed before, even in the case $k=1$. As one of many consequences, we give a short proof of the Stein-Weiss inequality in the context of $\mathbf R^{n-k} \times \mathbf R^n$.

math.FA↗
arXiv2023

BMO-type functionals, total variation, and $Γ$-convergence

We study the BMO-type functional $κ_{\varepsilon}(f,\mathbb R^n)$, which can be used to characterize BV functions $f\in BV(\mathbb R^n)$. The $Γ$-limit of this functional, taken with respect to $L^1_{\mathrm{loc}}$-convergence, is known to be $\tfrac 14 |Df|(\mathbb R^n)$. We show that the $Γ$-limit with respect to $L^{\infty}_{\mathrm{loc}}$-convergence is \[ \tfrac 14 |D^a f|(\mathbb R^n)+\tfrac 14 |D^c f|(\mathbb R^n)+\tfrac 12 |D^j f|(\mathbb R^n), \] which agrees with the ``pointwise'' limit in the case of SBV functions.

math.AP↗
arXiv2022

Universal potential estimates for $1<p\leq 2-\frac{1}{n}$

We extend the so-called universal potential estimates of the Kuusi-Mingione type (J.Funct. Anal. 2012) to the singular case $1<p\leq 2-1/n$ for the quasilinear equation with measure data \begin{equation*} -\operatorname{div}(A(x,\nabla u))=μ \end{equation*} in a bounded open subset $Ω$ of $\mathbb{R}^n$, $n\geq 2$, with a finite signed measure $μ$ in $Ω$. The operator $\operatorname{div}(A(x,\nabla u))$ is modeled after the $p$-Laplacian $Δ_p u:= {\rm div}\, (|\nabla u|^{p-2}\nabla u)$, where the nonlinearity $A(x, ξ)$ ($x, ξ\in \mathbb{R}^n$) is assumed to satisfy natural growth and monotonicity conditions of order $p$, as well as certain additional regularity conditions in the $x$-variable.

math.AP↗
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.

math.AP↗
arXiv2022

Sharp estimates for screened Vlasov-Poisson system around Penrose-stable equilibria in $\mathbb{R}^d $, $ d\geq3$

In this paper, we study the asymptotic stability of Penrose-stable equilibria among solutions of the screened Vlasov-Poisson system in $\mathbb{R}^d$ with $d\geq 3$ that was first established by Bedrossian, Masmoudi, and Mouhot in \cite{JBedrossian2018} with smooth initial data. More precisely, we prove the sharp decay estimates for the density of the perturbed system, exactly like the free transport with only Hölder (i.e., $C^{a}$ for $0<a<1$) perturbed initial data. This improves the recent works in \cite{HanKwanD2021} by Han-Kwan, Nguyen, and Rousset for lower derivatives of the density and in \cite{NguyenTT2020} by T. Nguyen for higher derivatives with a logarithmic correction in time. Furthermore, we establish new estimates and cancellations of the kernel to the linearized problem to obtain this result. Moreover, we also prove this result for the Vlasov-Poisson system in which the electric field obeys a general nonlinear Poisson equation containing massless electrons/ions case.

math.AP↗
arXiv2022

A comparison estimate for singular $p$-Laplace equations and its consequences

Comparison estimates are an important technical device in the study of regularity problems for quasilinear possibly degenerate elliptic and parabolic equations. Such tools have been employed indispensably in many papers of Mingione, Duzaar-Mingione, and Kuusi-Mingione, etc. on certain measure datum problems to obtain pointwise bounds for solutions and their full or fractional derivatives in terms of appropriate linear or nonlinear potentials. However, a comparison estimate for $p$-Laplace type elliptic equations with measure data is still unavailable in the strongly singular case $1< p\leq \frac{3n-2}{2n-1}$, where $n\geq 2$ is the dimension of the ambient space. This issue will be completely resolved in this work by proving a comparison estimate in a slightly larger range $1<p<3/2$. Applications include a `sublinear' Poincaré type inequality, pointwise bounds for solutions and their derivatives by Wolff's and Riesz's potentials, respectively. Some global pointwise and weighted estimates are also obtained for bounded domains, which enable us to treat a quasilinear Riccati type equation with possibly sublinear growth in the gradient.

math.AP↗
arXiv2022

On the dynamics of the roots of polynomials under differentiation

This article is devoted to the study of a nonlinear and nonlocal parabolic equation introduced by Stefan Steinerberger to study the roots of polynomials under differentiation; it also appeared in a work by Dimitri Shlyakhtenko and Terence Tao on free convolution. Rafael Granero-Belinchón obtained a global well-posedness result for positive initial data small enough in a Wiener space, and recently Alexander Kiselev and Changhui Tan proved a global well-posedness result for any positive initial data in the Sobolev space $H^s(\mathbb{S})$ with $s>3/2$. In this paper, we consider the Cauchy problem in the critical space $H^{1/2}(\mathbb{S})$. Two interesting new features, at this level of regularity, are that the equation can be written in the form $$ \partial_t u+V\partial_x u+γΛu=0, $$ where $γ$ is non-negative but not bounded from below and $V/\sqrtγ$ is not bounded. Therefore, the equation is only weakly parabolic. We prove that nevertheless the Cauchy problem is well posed locally in time and that the solutions are smooth for positive times. Combining this with the results of Kiselev and Tan, this gives a global well-posedness result for any positive initial data in $H^{1/2}(\mathbb{S})$. Our proof relies on sharp commutator estimates and introduces a strategy to prove a local well-posedness result in a situation where the lifespan depends on the profile of the initial data and not only on its norm.

math.AP↗
arXiv2021

The Muskat problem with $C^1$ data

In this paper we prove that the Cauchy problem of the Muskat equation is wellposed locally in time for any initial data in $\dot C^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$.

math.AP↗
arXiv2021

The Peskin problem with $\dot B^1_{\infty,\infty}$ initial data

In this paper we study the Peskin problem in 2D, which describes the dynamics of a 1D closed elastic structure immersed in a steady Stokes flow. We prove the local well-posedness for arbitrary initial configuration in $(C^2)^{\dot B^1_{\infty,\infty}}$ satisfying the well-stretched condition, and the global well-posedness when the initial configuration is sufficiently close to an equilibrium in $\dot B^1_{\infty,\infty}$. Here $(C^2)^{\dot B^1_{\infty,\infty}}$ is the closure of $C^2$ in the Besov space $\dot B^1_{\infty,\infty}$. The global-in-time solution will converge to an equilibrium exponentially as $t\rightarrow+\infty$. This is the first well-posedness result for the Peskin problem with non-Lipschitz initial data.

math.AP↗
arXiv2021

Hölder continuity of solutions for a class of drift-diffusion equations

We provide several regularity results for non-homogeneous drift-diffusion equations with applications to general dissipative SQG. Our results unify in a rather simple way several previously known results. We build the estimates on an algebraic identity (for the refined energy argument) which relates any integral operator with pure powers of the laplacian.

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