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Nguyen Anh Dao

Publications and source records attributed to Nguyen Anh Dao.

7 recordsLinked to original sources
arXiv2024

Sobolev interpolation inequalities with optimal Hardy-Rellich inequalities and critical exponents

We establish a new family of the critical higher order Sobolev interpolation inequalities for radial functions as well as for non-radial functions. These Sobolev interpolation inequalities are sharp in the sense that they use the optimal quadratic forms of the sharp Hardy-Rellich inequalities and cover the Sobolev critical exponents. Our results extend those studied by Dietze and Nam in [15] for the first order derivative case to higher order setting. The well-known Pólya-Szegö symmetrization principle and the nonlinear ground state representation play an important role in the work of [15]. To overcome the absence of the Pólya-Szegö principle and the nonlinear ground state representation in the higher order case, our proofs rely on the Fourier analysis and a higher order verion of the Talenti comparison principle. We also study a new version of the critical Hardy-Sobolev interpolation inequality involving the critical quadratic form of the Hardy inequality and Lorentz norms. Our critical Hardy-Sobolev interpolation inequality complements the result of Dietze and Nam in [15].

math.AP↗
arXiv2023

Decay estimates for time-fractional porous medium flow with nonlocal pressure

The main purpose of this paper is to study weak solutions of time-fractional of porous medium equation with nonlocal pressure: \[ \partial^α_t u=\operatorname{div}\left( |u|^{m}\nabla (-Δ)^{-s} u\right) \,\, \text{in } \mathbb{R}^N\times (0,T) \,, \] with $m\geq 1$, $N\geq 2$, $\frac{1}{2}\leq s<1$, and $α\in(0,1)$. We first prove an existence of weak solutions to the equation with initial data in $L^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ (possibly mixed sign). After that, we establish the $L^q-L^\infty$ decay estimate of weak solutions: \[ \|u(t)\|_{L^\infty(\mathbb{R}^N)} \leq C t^{-\fracα{q(1-λ_0)+ m}} \|u_0\|_{L^{q}(\mathbb{R}^N)}^{\frac{q(1-λ_0)}{q(1-λ_0) + m}} ,\quad \text{for } t\in(0,\infty), \] with $λ_0=\frac{N-2(1-s)}{N}$.

math.AP↗
arXiv2020

Pointwise gradient estimates in multi-dimensional slow diffusion equations with a singular quenching term

We consider a slow diffusion equation with a singular quenching term, extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner for the one-dimensional case. Besides the existence of a very weak solution, we prove some pointwise gradient estimates, mainly when the absorption dominates over the diffusion. In particular, a new kind of universal gradient estimate is proved. Several qualitative properties (such as the finite time quenching phenomena and the finite speed of propagation) and the study of the Cauchy problem are also considered.

math.AP↗
arXiv2015

Initial trace of solutions of Hamilton-Jacobi parabolic equation with absorption

Here we study the initial trace problem for the nonnegative solutions of the equation \[ u\_{t}-Δu+|\nabla u|^{q}=0 \] in $Q\_{Ω,T}=Ω\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q>0,$ and $Ω=\mathbb{R}^{N},$ or $Ω$ is a smooth bounded domain of $\mathbb{R}^{N}$ and $u=0$ on $\partialΩ\times\left( 0,T\right) .$ We can define the trace at $t=0$ as a nonnegative Borel measure $(\mathcal{S} ,u\_{0}),$ where $S$ is the closed set where it is infinite, and $u\_{0}$ is a Radon measure on $Ω\backslash\mathcal{S}.$ We show that the trace is a Radon measure when $q\leqq1.$ For $q\in(1,(N+2)/(N+1)$ and any given Borel measure, we show the existence of a minimal solution, and a maximal one on conditions on $u\_{0}.$ When $\mathcal{S}$ $=\overlineω\capΩ$ and $ω$ is an open subset of $Ω,$ the existence extends to any $q\leqq2$ when $u\_{0}\in L\_{loc}^{1}(Ω)$ and any $q>1$ when $u\_{0}=0$. In particular there exists a self-similar nonradial solution with trace $(\mathbb{R}^{N+},0),$ with a growth rate of order $\left\vert x\right\vert ^{q^{\prime}}$ as $\left\vert x\right\vert \rightarrow\infty$ for fixed $t.$ Moreover we show that the solutions with trace $(\overlineω,0)$ in $Q\_{\mathbb{R}^{N},T}$ may present near $t=0$ a growth rate of order $t^{-1/(q-1)}$ in $ω$ and of order $t^{-(2-q)/(q-1)}$ on $\partial ω.$

math.AP↗
arXiv2013

$L^{\infty}$ estimates and uniqueness results for nonlinear parabolic equations with gradient absorption terms

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem \[ \left\{\begin{array} [c]{c}% u_{t}-νΔu+|\nabla u|^{q}=0, u(0)=u_{0}, \end{array} \right. \] in $Q_{Ω,T}=Ω\times\left(0,T\right) ,$ where $q>1,ν\geqq 0,T\in\left(0,\infty\right] ,$ and $Ω=\mathbb{R}^{N}$ or $Ω$ is a smooth bounded domain, and $u_{0}\in L^{r}(Ω),r\geqq1,$ or $u_{0}% \in\mathcal{M}_{b}(Ω).$ We show $L^{\infty}$ decay estimates, valid for \textit{any weak solution}, \textit{without any conditions a}s $\left\| x\right\| \rightarrow\infty,$ and \textit{without uniqueness assumptions}. As a consequence we obtain new uniqueness results, when $u_{0}\in \mathcal{M}_{b}(Ω)$ and $q<(N+2)/(N+1),$ or $u_{0}\in L^{r}(Ω)$ and $q<(N+2r)/(N+r).$ We also extend some decay properties to quasilinear equations of the model type \[ u_{t}-Δ_{p}u+\left\| u\right\| ^{λ-1}u|\nabla u|^{q}=0 \] where $p>1,λ\geqq0,$ and $u$ is a signed solution.

math.AP↗
arXiv2012

Isolated initial singularities for the viscous Hamilton-Jacobi equation

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi equation [u_{t}-Δu+|\nabla u|^{q}=0] in $Q_{Ω,T}=Ω\times(0,T),$ where $q>1,T\in(0,\infty] ,$ and $Ω$ is a smooth bounded domain of $\mathbb{R}% ^{N}$ containing $0,$ or $Ω=\mathbb{R}^{N}.$ We consider solutions with a possible singularity at point $(x,t)=(0,0).$ We show that if $q\geq q_{\ast}=(N+2)/(N+1)$ the singularity is removable.

math.AP↗