arXiv subjects

Phuoc-Tai Nguyen

Publications and source records attributed to Phuoc-Tai Nguyen.

At least 19 recordsLinked to original sources

arXiv2026

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.

math.AP↗

arXiv2026

Nonlinear Schrödinger equations with critical Hardy potential and Choquard nonlinearity

We study the Cauchy problem for the nonlinear Schrödinger equation characterized by contrasting effects between the concentration at the origin of a critical Hardy potential and the intrinsic nonlocality of a Choquard nonlinearity. We prove the existence of a ground state solution through optimizers of an interpolation Hardy-Gagliardo-Nirenberg inequality and derive a non-existence result via Poho\v zaev identities. Using these results, we provide various criteria for the global existence and finite-time blow-up for the problem in the energy-subcritical regime. Finally, we establish a key compactness result, which enables us to obtain a characterization of finite-time blow-up solutions with minimal mass.

math.AP↗

arXiv2026

Polynomial Null Solutions to Bosonic Laplacians, Bosonic Bergman and Hardy Spaces

A bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.

math.CV↗

arXiv2025

Finite-Rank Optimizers for the mass--supercritical Lieb--Thirring and Hardy--Lieb--Thirring Inequalities

We establish the existence of finite-rank operators for an interpolation version of the Lieb--Thirring inequality in the mass--supercritical case, thereby extending a result of Hong, Kwon, and Yoon in 2019 to the full parameter regime. Our method also applies to the Hardy--Lieb--Thirring inequality, where the existence of optimizers faces additional difficulties due to the singularity of the inverse-square potential.

math-ph↗

arXiv2025

Well-Posedness for Fractional Reaction-Diffusion Systems with Mass Dissipation in $\mathbb R^N$

The global existence of bounded solutions to reaction-diffusion systems with fractional diffusion in the whole space $\mathbb R^N$ is investigated. The systems are assumed to preserve the non-negativity of initial data and to dissipate total mass. We first show that if the nonlinearities are at most quadratic then there exists a unique global bounded solution regardless of the fractional order. This is done by combining a regularizing effect of the fractional diffusion operator and the Hölder continuity of a non-local inhomogeneous parabolic equation. When the nonlinearities might be super-quadratic, but satisfy some intermediate sum conditions, we prove the global existence of bounded solutions by adapting the well-known duality methods to the case of fractional diffusion. In this case, the order of the intermediate sum conditions depends on the fractional order. These results extend the existing theory for mass dissipated reaction-diffusion systems to the case of non-local diffusion and unbounded domains.

math.AP↗

arXiv2025

Elliptic Schrödinger equations with gradient-dependent nonlinearity and Hardy potential singular on manifolds

Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $Σ\subset Ω$ is a $C^2$ compact boundaryless submanifold in $\mathbb{R}^N$ of dimension $k$, $0\leq k < N-2$. For $μ\leq (\frac{N-k-2}{2})^2$, put $L_μ:= Δ+ μd_Σ^{-2}$ where $d_Σ(x) = \mathrm{dist}(x,Σ)$. We study boundary value problems for equation $-L_μu = g(u,|\nabla u|)$ in $Ω\setminus Σ$, subject to the boundary condition $u=ν$ on $\partial Ω\cup Σ$, where $g: \mathbb{R} \times \mathbb{R}_+ \to \mathbb{R}_+$ is a continuous and nondecreasing function with $g(0,0)=0$, $ν$ is a given nonnegative measure on $\partial Ω\cup Σ$. When $g$ satisfies a so-called subcritical integral condition, we establish an existence result for the problem under a smallness assumption on $ν$. If $g(u,|\nabla u|) = |u|^p|\nabla u|^q$, there are ranges of $p,q$, called subcritical ranges, for which the subcritical integral condition is satisfied, hence the problem admits a solution. Beyond these ranges, where the subcritical integral condition may be violated, we establish various criteria on $ν$ for the existence of a solution to the problem expressed in terms of appropriate Bessel capacities.

math.AP↗

arXiv2024

Compactness of Green operators with applications to semilinear nonlocal elliptic equations

In this paper, we consider a class of integro-differential operators $\mathbb{L}$ posed on a $C^2$ bounded domain $Ω\subset \mathbb{R}^N$ with appropriate homogeneous Dirichlet conditions where each of which admits an inverse operator commonly known as the Green operator $\mathbb{G}^Ω$. Under mild conditions on $\mathbb{L}$ and its Green operator, we establish various sharp compactness of $\mathbb{G}^Ω$ involving weighted Lebesgue spaces and weighted measure spaces. These results are then employed to prove the solvability for semilinear elliptic equation $\mathbb{L} u + g(u) = μ$ in $Ω$ with boundary condition $u=0$ on $\partial Ω$ or exterior condition $u=0$ in $\mathbb{R}^N \setminus Ω$ if applicable, where $μ$ is a Radon measure on $Ω$ and $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing continuous function satisfying a subcriticality integral condition. When $g(t)=|t|^{p-1}t$ with $p>1$, we provide a sharp sufficient condition expressed in terms of suitable Bessel capacities for the existence of a solution. The contribution of the paper consists of (i) developing novel unified techniques which allow to treat various types of fractional operators and (ii) obtaining sharp compactness and existence results in weighted spaces, which refine and extend several related results in the literature.

math.AP↗

arXiv2024

Cwikel-Lieb-Rozenblum type inequalities for Hardy-Schrödinger operator

We prove a Cwikel-Lieb-Rozenblum type inequality for the number of negative eigenvalues of the Hardy-Schrödinger operator $-Δ- (d-2)^2/(4|x|^2) -W(x)$ on $L^2(\mathbb{R}^d)$. The bound is given in terms of a weighted $L^{d/2}-$norm of $W$ which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian.

math-ph↗

arXiv2024

Semiclassical Moser-Trudinger inequalities

We extend the Moser-Trudinger inequality of one function to systems of orthogonal functions. Our results are asymptotically sharp when applied to the collective behavior of eigenfunctions of Schrödinger operators on bounded domains.

math.AP↗

arXiv2023

Boundary value problems for semilinear Schrödinger equations with singular potentials and measure data

We study boundary value problems with measure data in smooth bounded domains $Ω$, for semilinear equations involving Hardy type potentials. Specifically we consider problems of the form $-L_V u + f(u) = τ$ in $Ω$ and $\mathrm{tr}^*u=ν$ on $\partial Ω$, where $L_V= Δ+V$, $f\in C(\mathbb{R})$ is monotone increasing with $f(0)=0$ and $\mathrm{tr}^*u$ denotes the normalized boundary trace of $u$ associated with $L_V$. The potential $V$ is typically a Hölder continuous function in $Ω$ that explodes as $\mathrm{dist}(x,F)^{-2}$ for some $F \subset \partial Ω$. In general the above boundary value problem may not have a solution. We are interested in questions related to the concept of 'reduced measures', introduced by Brezis, Marcus and Ponce for $V=0$. For positive measures, the reduced measures $τ^*, ν^*$ are the largest measures dominated by $τ$ and $ν$ respectively such that the boundary value problem with data $(τ^*,ν^*)$ has a solution. Our results extend results for the case $V=0$, including a relaxation of the conditions on $f$. In the case of signed measures, some of the present results are new even for $V=0$.

math.AP↗

arXiv2023

Semilinear elliptic equations involving fractional Hardy operators

Our aim in this article is to study semilinear elliptic equations involving a fractional Hardy operator, an absorption and a Radon source in a weighted distributional sense. We show various scenarios, produced by the combined effect of the fractional Hardy potential, the growth of the absorption term and the concentration of the measure, in which existence and uniqueness results holds.

math.AP↗

arXiv2022

A large class of nonlocal elliptic equations with singular nonlinearities

In this work, we address the questions of existence, uniqueness, and boundary behavior of the positive weak-dual solution of equation $\mathbb{L}_γ^s u = \mathcal{F}(u)$, posed in a $C^2$ bounded domain $Ω\subset \mathbb{R}^N$, with appropriate homogeneous boundary or exterior Dirichlet conditions. The operator $\mathbb{L}_γ^s$ belongs to a general class of nonlocal operators including typical fractional Laplacians such as restricted fractional Laplacian, censored fractional Laplacian and spectral fractional Laplacian. The nonlinear term $\mathcal{F}(u)$ covers three different amalgamation of nonlinearities: a purely singular nonlinearity $\mathcal{F}(u) = u^{-q}$ ($q>0$), a singular nonlinearity with a source term $\mathcal{F}(u) = u^{-q} + f(u)$, and a singular nonlinearity with an absorption term $\mathcal{F}(u) = u^{-q}-g(u)$. Based on a delicate analysis of the Green kernel associated to $\mathbb{L}_γ^s$, we develop a new unifying approach that empowered us to construct a theory for equation $\mathbb{L}_γ^s u = \mathcal{F}(u)$. In particular, we show the existence of two critical exponents $q^{\ast}_{s, γ}$ and $q^{\ast \ast}_{s, γ}$ which provides a fairly complete classification of the weak-dual solutions via their boundary behavior. Various types of nonlocal operators are discussed to exemplify the wide applicability of our theory.

math.AP↗

arXiv2022

Semilinear elliptic equations involving power nonlinearities and hardy potentials with boundary singularities

Let $Ω\subset\mathbb{R}^N$ ($N\geq 3$) be a $C^2$ bounded domain and $Σ\subset \partialΩ$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0\leq k \leq N-1$. We assume that $Σ= \{0\}$ if $k = 0$ and $Σ=\partialΩ$ if $k=N-1$. Denote $d_Σ(x)=\mathrm{dist}(x,Σ)$ and put $L_μ=Δ+ μd_Σ^{-2}$ where $μ$ is a parameter. In this paper, we study boundary value problems for equations $-L_μu \pm |u|^{p-1}u = 0$ in $Ω$ with prescribed condition $u=ν$ on $\partial Ω$, where $p>1$ and $ν$ is a given measure on $\partial Ω$. The nonlinearity $|u|^{p-1}u$ is referred to as \textit{absorption} or \textit{source} depending whether the plus sign or minus sign appears. The distinctive feature of the problems is characterized by the interplay between the concentration of $Σ$, the type of nonlinearity, the exponent $p$ and the parameter $μ$. The absorption case and the source case are sharply different in several aspects and hence require completely different approaches. In each case, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities. In comparison with related works in the literature, by employing a fine analysis, we are able to treat the supercritical ranges for the exponent $p$, and the critical case for the parameter $μ$, which justifies the novelty of our paper.

math.AP↗

arXiv2022

Semilinear nonlocal elliptic equations with source term and measure data

Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) ${\mathbb L} u = u^p +λμ$ in a bounded domain $Ω$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $λ>0$. The operator ${\mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $μ$ is taken in the optimal weighted measure space. The interplay between the operator ${\mathbb L}$, the source term $u^p$ and the datum $μ$ yields substantial difficulties and reveals the distinctive feature of the problem. We develop a new unifying technique based on a fine analysis on the Green kernel, which enables us to construct a theory for semilinear equation (E) in measure frameworks. A main thrust of the paper is to provide a fairly complete description of positive solutions to the Dirichlet problem for (E). In particular, we show that there exist a critical exponent $p^*$ and a threshold value $λ^*$ such that the multiplicity holds for $1<p<p^*$ and $0<λ<λ^*$, the uniqueness holds for $1<p<p^*$ and $λ=λ^*$, and the nonexistence holds in other cases. Various types of nonlocal operator are discussed to exemplify the wide applicability of our theory.

math.AP↗

arXiv2022

Semilinear elliptic Schrödinger equations involving singular potentials and source terms

Let $Ω\subset \mathbb{R}^N$ ($N>2$) be a $C^2$ bounded domain and $Σ\subset Ω$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_μ= Δ+ μd_Σ^{-2}$ in $Ω\setminus Σ$, where $d_Σ(x) = \mathrm{dist}(x,Σ)$ and $μ$ is a parameter. We study the boundary value problem (P) $-L_μu = g(u) + τ$ in $Ω\setminus Σ$ with condition $u=ν$ on $\partial Ω\cup Σ$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function and $τ$ and $ν$ are positive measures. The interplay between the inverse-square potential $d_Σ^{-2}$, the nature of the source term $g(u)$ and the measure data $τ,ν$ yields substantial difficulties in the research of the problem. We perform a deep analysis based on delicate estimate on the Green kernel and Martin kernel and fine topologies induced by appropriate capacities to establish various necessary and sufficient conditions for the existence of a solution in different cases.

math.AP↗

arXiv2022

Semilinear elliptic Schrödinger equations with singular potentials and absorption terms

Let $Ω\subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $Σ\subset Ω$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_μ= Δ+ μd_Σ^{-2}$ in $Ω\setminus Σ$, where $d_Σ(x) = \mathrm{dist}(x,Σ)$ and $μ$ is a parameter. We investigate the boundary value problem (P) $-L_μu + g(u) = τ$ in $Ω\setminus Σ$ with condition $u=ν$ on $\partial Ω\cup Σ$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function, and $τ$ and $ν$ are positive measures. The complex interplay between the competing effects of the inverse-square potential $d_Σ^{-2}$, the absorption term $g(u)$ and the measure data $τ,ν$ discloses different scenarios in which problem (P) is solvable. We provide sharp conditions on the growth of $g$ for the existence of solutions. When $g$ is a power function, namely $g(u)=|u|^{p-1}u$ with $p>1$, we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e. $p$ is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e. $p$ is greater than a critical exponent). Finally, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities for the solvability of (P).

math.AP↗

arXiv2020

Boundary value problem with measures for fractional elliptic equations involving source nonlinearities

We are concerned with positive solutions of equation (E) $(-Δ)^s u=f(u)$ in a domain $Ω\subset \mathbb{R}^N$ ($N>2s$), where $s \in (\frac{1}{2},1)$ and $f\in C^α_{loc}(\mathbb{R})$ for some $α\in(0,1)$. We establish a universal a priori estimate for positive solutions of (E), as well as for their gradients. Then for $C^2$ bounded domain $Ω$, we prove the existence of positive solutions of (E) with prescribed boundary value $ρν$, where $ρ>0$ and $ν$ is a positive Radon measure on $\partial Ω$ with total mass $1$, and discuss regularity property of the solutions. When $f(u)=u^p$, we demonstrate that there exists a critical exponent $p_s:=\frac{N+s}{N-s}$ in the following sense. If $p\geq p_s$, the problem does not admit any positive solution with $ν$ being a Dirac mass. If $p\in(1,p_s)$ there exits a threshold value $ρ^*>0$ such that for $ρ\in (0, ρ^*]$, the problem admits a positive solution and for $ρ>ρ^*$, no positive solution exists. We also show that, for $ρ>0$ small enough, the problem admits at least two positive solutions.

math.AP↗

arXiv2020

Boundary value problems in Euclidean space for Bosonic Laplacians

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with $L^p$ boundary data, $1 \leq p \leq \infty$. We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc.

math-ph↗