arXiv subjects

Tuoc Phan

Publications and source records attributed to Tuoc Phan.

At least 19 recordsLinked to original sources

arXiv2026

Weighted $W^{1,p}$-estimates for Parabolic Equations of Fabes-Kenig-Seraponi singular-degenerate type

We investigate Dirichlet boundary value problems for a class of second-order parabolic equations in divergence-form with coefficient matrices that exhibit singular and degenerate behaviors characterized by a Muckenhoupt weight class. This framework serves as the parabolic analogue to the singular-degenerate elliptic equations pioneered by Fabes, Kenig, and Seraponi. Under a smallness assumption on the partially weighted mean oscillation of the coefficients, we establish the existence, uniqueness, and local interior and boundary regularity estimates for weak solutions within appropriately defined weighted Sobolev spaces. The proofs rely on the freezing coefficient technique alongside the level-set method introduced by Caffarelli and Peral. Additionally, we develop the necessary weighted Sobolev space framework and related weighted inequalities. Finally, a compactness argument is utilized to demonstrate that solutions to these equations remain locally close, in the weighted Sobolev norm, to their frozen-coefficient counterparts.

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arXiv2026

On $W^{2,\varepsilon}$-estimates for a class of singular-degenerate parabolic equations

We study a class of parabolic equations in non-divergence form with measurable coefficients that exhibit singular and/or degenerate behavior governed by weights in the $A_{1+\frac{1}{n}}$-Muckenhoupt class. Under a smallness assumption on a weighted mean oscillation of the weights, we establish weighted $W^{2,\varepsilon}$-estimates in the spirit of F.-H. Lin. Our results particularly holds for equations whose leading coefficients are of logistic-type singularities, as well as to those with polynomial blow-up or vanishing with sufficiently small exponents. A central component of our approach is the development of local quantitative lower estimates for solutions, which are interpreted as the mean sojourn time of sample paths, a stochastic-geometric perspective that generalizes the seminal work of L. C. Evans. We address the singular-degenerate nature of the operators by employing a class of intrinsic weighted parabolic cylinders, combined with a perturbation argument and parabolic Aleksandrov-Bakelman-Pucci (ABP) estimates. Furthermore, we conduct a rigorous analysis of weight regularization and truncation to ensure that the estimates are independent of the regularization and truncation parameters. The results extend classical regularity theory to a broad class of second-order parabolic equations and provide a functional analytic foundation for further study of fully nonlinear parabolic equations with singular-degenerate structure

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arXiv2025

Well-posedness for a class of parabolic equations with singular-degenerate coefficients

This paper studies a class of linear parabolic equations with measurable coefficients in divergence form whose volumetric heat capacity coefficients are assumed to be in some Muckenhoupt class of weights. As such, the coefficients can be degenerate, singular, or both degenerate and singular. A class of weighted parabolic cylinders with a non-homogeneous quasi-distance function, and a class of weighted parabolic Sobolev spaces intrinsically suitable for the class of equations are introduced. Under some smallness assumptions on the mean oscillations of the coefficients, regularity estimates, existence, and uniqueness of weak solutions in the weighted Sobolev spaces are proved. To achieve the results, we apply the level-set method introduced by Caffarelli and Peral. Several weighted inequalities and a weighted Aubin-Lions compactness theorem for sequences in weighted parabolic Sobolev spaces are established.

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arXiv2025

An inviscid limit problem for Navier-Stokes equations in 3D domains with oscillatory boundaries

We study an inviscid limit problem for a class of Navier-Stokes equations with vanishing measurable viscous coefficients in 3-dimensional spatial domains whose boundaries are oscillatory, depending on a small parameter, and become flat when the parameter converges to zero. Under some sufficient conditions on the anisotropic vanishing rates of the eigenvalues of the matrices of the viscous coefficients and the oscillatory parameter, we show that Leray-Hopf weak solutions of the Navier-Stokes equations with no slip boundary condition converge to solutions of the Euler equations in the upper half space. To prove the result, we apply a change of variables to flatten the boundaries of the spatial domains for the Navier-Stokes equations, and then construct the boundary layer terms. As the Navier-Stokes equations and the Euler equations are originally written in two different domains, additional boundary layer terms are constructed and their estimates are obtained.

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arXiv2024

Harnack inequality for singular or degenerate parabolic equations in non-divergence form

This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\frac{1}{n}}$ class of Muckenhoupt weights. Krylov-Safonov Harnack inequality for solutions is proved under some smallness assumption on a weighted mean oscillation of the weight. To prove the result, we introduce a class of generic weighted parabolic cylinders and the smallness condition on the weighted mean oscillation of the weight through which several growth lemmas are established. Additionally, a perturbation method is used and the parabolic Aleksandrov-Bakelman-Pucci type maximum principle is crucially applied to suitable barrier functions to control the solutions. As corollaries, H\"{o}lder regularity estimates of solutions with respect to a quasi-distance, and a Liouville type theorem are obtained in the paper.

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arXiv2023

On linear elliptic equations with drift terms in critical weak spaces

We study the Dirichlet problem for a second order linear elliptic equation in a bounded smooth domain $Ω$ in $\mathbb{R}^n$, $n \ge 3$, with the drift $\mathbf{b} $ belonging to the critical weak space $L^{n,\infty}(Ω)$. We decompose the drift $\mathbf{b} = \mathbf{b}_1 + \mathbf{b}_2$ in which $\text{div} \mathbf{b}_1 \geq 0$ and $\mathbf{b}_2$ is small only in a small scale quasi-norm of $L^{n,\infty}(Ω)$. Under this new smallness condition, we prove existence, uniqueness, and regularity estimates of weak solutions to the problem and its dual. Hölder regularity and derivative estimates of weak solutions to the dual problem are also established. As a result, we prove uniqueness of very weak solutions slightly below the threshold. When $\mathbf{b}_2 =0$, our results recover those by Kim and Tsai in [SIAM J. Math. Anal. 52 (2020)]. Due to the new small scale quasi-norm, our results are new even when $\mathbf{b}_1=0$.

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arXiv2023

Nondivergence form degenerate linear parabolic equations on the upper half space

We study a class of nondivergence form second-order degenerate linear parabolic equations in $(-\infty, T) \times {\mathbb R}^d_+$ with the homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial {\mathbb R}^d_+$, where ${\mathbb R}^d_+ = \{x =(x_1,x_2,\ldots, x_d) \in {\mathbb R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The coefficient matrices of the equations are the product of $μ(x_d)$ and bounded positive definite matrices, where $μ(x_d)$ behaves like $x_d^α$ for some given $α\in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain. The divergence form equations in this setting were studied in [14]. Under a partially weighted VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our research program is motivated by the regularity theory of solutions to degenerate viscous Hamilton-Jacobi equations.

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arXiv2023

On stationary Navier-Stokes equations in the upper-half plane

We study the incompressible stationary Navier-Stokes equations in the upper-half plane with homogeneous Dirichlet boundary condition and non-zero external forcing terms. Existence of weak solutions is proved under a suitable condition on the external forces. Weak-strong uniqueness criteria based on various growth conditions at the infinity of weak solutions are also given. This is done by employing an energy estimate and a Hardy's inequality. Several estimates of stream functions are carried out and two density lemmas with suitable weights for the homogeneous Sobolev space on 2 dimensional space are proved.

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arXiv2023

Sobolev estimates for singular-degenerate quasilinear equations beyond the $A_2$ class

We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain $Ω$ whose coefficients can be degenerate or singular of the type $\text{dist}(x, \partial Ω)^α$, where $\partial Ω$ is the boundary of $Ω$ and $α\in (-1, \infty)$ is a given number. We establish weighted Sobolev type estimates for weak solutions under a smallness assumption on the weighted mean oscillations of the coefficients in small balls. Our approach relies on a perturbative method and several new Lipschitz estimates for weak solutions to a class of singular-degenerate quasilinear equations.

math.AP↗

arXiv2022

On trace Theorems for weighted mixed norm Sobolev spaces and applications

We prove trace theorems for weighted mixed norm Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the power in the weight function and the integrability power for the integration with respect to the vertical variable but not on the integrability powers for the integration with respect to the horizontal ones. They are new even in the un-weighted case and they recover classical results in the case of un-mixed norm spaces. The work is motivated by the study of regularity theory for solutions of elliptic and parabolic equations with anisotropic features and with non-homogeneous boundary conditions. The results provide an essential ingredient to the study of fractional elliptic and parabolic equations in divergence form with measurable coefficients.

math.AP↗

arXiv2022

Weighted mixed-norm $L_p$ estimates for equations in non-divergence form with singular coefficients: the Dirichlet problem

We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the equations under consideration appear in the study of fractional heat and fractional Laplace equations. Intrinsic weighted Sobolev spaces are found in which the existence and uniqueness of strong solutions are proved under certain smallness conditions on the weighted mean oscillations of the coefficients in small parabolic cylinders. Our results are new even when the coefficients are constants and they cover the case where the weights may not be in the $A_p$-Muckenhoupt class.

math.AP↗

arXiv2022

Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients

We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm $L_{s,q}$-estimates hold for the spatial second-order derivatives of solutions, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with $s=q=2$, the result is still new and provides local boundary Caccioppoli-type estimates. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for both divergence and non-divergence form parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.

math.AP↗

arXiv2021

Degenerate linear parabolic equations in divergence form on the upper half space

We study a class of second-order degenerate linear parabolic equations in divergence form in $(-\infty, T) \times \mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial \mathbb R^d_+$, where $\mathbb R^d_+ = \{x \in \mathbb R^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given. The coefficient matrices of the equations are the product of $μ(x_d)$ and bounded uniformly elliptic matrices, where $μ(x_d)$ behaves like $x_d^α$ for some given $α\in (0,2)$, which are degenerate on the boundary $\{x_d=0\}$ of the domain. Under a partially VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.

math.AP↗

arXiv2021

On a class of divergence form linear parabolic equations with degenerate coefficients

We study a class of linear parabolic equations in divergence form with degenerate coefficients on the upper half space. Specifically, the equations are considered in $(-\infty, T) \times \mathbb{R}^d_+$, where $\mathbb{R}^d_+ = \{x \in \mathbb{R}^d\,:\, x_d>0\}$ and $T\in {(-\infty, \infty]}$ is given, and the diffusion matrices are the product of $x_d$ and bounded uniformly elliptic matrices, which are degenerate at $\{x_d=0\}$. As such, our class of equations resembles well the corresponding class of degenerate viscous Hamilton-Jacobi equations. We obtain wellposedness results and regularity type estimates in some appropriate weighted Sobolev spaces for the solutions.

math.AP↗

arXiv2020

Parabolic and elliptic equations with singular or degenerate coefficients: the Dirichlet problem

We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^α, α\in (-\infty, 1),$ and a bounded uniformly elliptic matrix of coefficients. Thus, the coefficients are singular or degenerate near the boundary $\{x_d =0\}$ and they may not locally integrable. The novelty of the work is that we find proper weights under which the existence, uniqueness, and regularity of solutions in Sobolev spaces are established. These results appear to be the first of their kind and are new even if the coefficients are constant. They are also readily extended to systems of equations.

math.AP↗

arXiv2020

On parabolic and elliptic equations with singular or degenerate coefficients

We study both divergence and non-divergence form parabolic and elliptic equations in the half space $\{x_d>0\}$ whose coefficients are the product of $x_d^α$ and uniformly nondegenerate bounded measurable matrix-valued functions, where $α\in (-1, \infty)$. As such, the coefficients are singular or degenerate near the boundary of the half space. For equations with the conormal or Neumann boundary condition, we prove the existence, uniqueness, and regularity of solutions in weighted Sobolev spaces and mixed-norm weighted Sobolev spaces when the coefficients are only measurable in the $x_d$ direction and have small mean oscillation in the other directions in small cylinders. Our results are new even in the special case when the coefficients are constants, and they are reduced to the classical results when $α=0$

math.AP↗

arXiv2019

Well-posedness for the Navier-Stokes equations in critical mixed-norm Lebesgue spaces

We study the Cauchy problem in $n$-dimensional space for the system of Navier-Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical mixed-norm Lebesgue spaces. Being in the mixed-norm Lebesgue spaces, both of the initial data and the solutions could be singular at certain points or decaying to zero at infinity with different rates in different spatial variable directions. Some of these singular rates could be very strong and some of the decaying rates could be significantly slow. Besides other interests, the results of the paper particularly show an interesting phenomena on the persistence of the anisotropic behavior of the initial data under the evolution. To achieve the goals, fundamental analysis theory such as Young's inequality, time decaying of solutions for heat equations, the boundedness of the Helmholtz-Leray projection, and the boundedness of the Riesz tranfroms are developed in mixed-norm Lebesgue spaces. These fundamental analysis results are independently topics of great interests and they are potentially useful in other problems.

math.AP↗

arXiv2018

Liouville type theorems for 3D stationary Navier-Stokes equations in weighted mixed-norm Lebesgue spaces

This work studies the system of $3D$ stationary Navier-Stokes equations. Several Liouville type theorems are established for solutions in mixed-norm Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show that, under some sufficient conditions in mixed-norm Lebesgue spaces, solutions of the stationary Navier-Stokes equations are identically zero. This result covers the important case that solutions may decay to zero with different rates in different spatial directions, and some these rates could be significantly slow. In the un-mixed norm case, the result recovers available results. With some additional geometric assumptions on the supports of solutions, this work also provides several other important Liouville type theorems for solutions in weighted mixed-norm Lebesgue spaces. To prove the results, we establish some new results on mixed-norm and weighted mixed-norm estimates for Navier-Stokes equations. All of these results are new and could be useful in other studies.

math.AP↗