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G. K. Duong

Publications and source records attributed to G. K. Duong.

5 recordsLinked to original sources
arXiv2025

Lieb--Thirring inequalities for large quantum systems with inverse nearest-neighbor interactions

We prove an analogue of the Lieb--Thirring inequality for many-body quantum systems with the kinetic operator $\sum_i (-Δ_i)^s$ and the interaction potential of the form $\sum_i δ_i^{-2s}$ where $δ_i$ is the nearest-neighbor distance to the point $x_i$. Our result extends the standard Lieb--Thirring inequality for fermions and applies to quantum systems without the anti-symmetry assumption on the wave functions. Additionally, we derive similar results for the Hardy--Lieb--Thirring inequality and obtain the asymptotic behavior of the optimal constants in the strong coupling limit.

math-ph↗
arXiv2024

Non-self similar blowup solutions to the higher dimensional Yang-Mills heat flows

In this paper, we consider the Yang-Mills heat flow on $\mathbb R^d \times SO(d)$ with $d \ge 11$. Under a certain symmetry preserved by the flow, the Yang-Mills equation can be reduced to: $$ \partial_t u =\partial_r^2 u +\frac{d+1}{r} \partial_r u -3(d-2) u^2 - (d-2) r^2 u^3, \text{ and } (r,t) \in \mathbb R_+ \times \mathbb R_+. $$ We are interested in describing the singularity formation of this parabolic equation. We construct non-self-similar blowup solutions for $d \ge 11$ and prove that the asymptotic of the solution is of the form $$ u(r,t) \sim \frac{1}{λ_\ell(t)} \mathcal{Q} \left( \frac{r}{\sqrt{λ_\ell (t)}} \right), \text{ as } t \to T ,$$ where $\mathcal{Q}$ is the ground state with boundary conditions $\mathcal{Q}(0)=-1, \mathcal{Q}'(0)=0$ and the blowup speed $λ_\ell$ verifies $$λ_\ell (t) = \left( C(u_0) +o_{t\to T}(1) \right) (T-t)^{\frac{2\ell }α} \text{ as } t \to T,~~ \ell \in \mathbb{N}^*_+, ~~α>1.$$ In particular, when $\ell = 1$, this asymptotic is stable whereas for $ \ell \ge 2$ it becomes stable on a space of codimension $\ell-1$. Our approach here is not based on energy estimates but on a careful construction of time dependent eigenvectors and eigenvalues combined with maximum principle and semigroup pointwise estimates.

math.AP↗
arXiv2022

Sharp equivalent for the blowup profile to the gradient of a solution to the semilinear heat equation

In this paper, we consider the standard semilinear heat equation \begin{eqnarray*} \partial_t u = Δu + |u|^{p-1}u, \quad p >1. \end{eqnarray*} The determination of the (believed to be) generic blowup profile is well-established in the literature, with the solution blowing up only at one point. Though the blow-up of the gradient of the solution is a direct consequence of the single-point blow-up property and the mean value theorem, there is no determination of the final blowup profile for the gradient in the literature, up to our knowledge. In this paper, we refine the construction technique of Bricmont-Kupiainen 1994 and Merle-Zaag 1997, and derive the following profile for the gradient: %and derive construct a blowup solution to the above equation with the gradient's asymptotic $$ \nabla u(x,T) \sim - \frac{\sqrt{2b}}{p-1} \frac{x}{|x| \sqrt{ |\ln|x||}} \left[\frac{b|x|^2}{2|\ln|x||} \right]^{-\frac{p+1}{2(p-1)}} \text{ as } x \to 0, $$ where $ b =\frac{(p-1)^2}{4p}$, which is as expected the gradient of the well-known blowup profile of the solution.

math.AP↗
arXiv2017

Construction of a stable blowup solution with a prescribed behavior for a non-scaling invariant semilinear heat equation

We consider the semilinear heat equation \begin{eqnarray*} \partial_t u = Δu + |u|^{p-1} u \ln ^α( u^2 +2), \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $p > 1$ and $ α\in \mathbb{R}$. Unlike the standard case $α= 0$, this equation is not scaling invariant. We construct for this equation a solution which blows up in finite time $T$ only at one blowup point $a$, according to the following asymptotic dynamics: \begin{eqnarray*} u(x,t) \sim ψ(t) \left(1 + \frac{(p-1)|x-a|^2}{4p(T -t)|\ln(T -t)|} \right)^{-\frac{1}{p-1}} \text{ as } t \to T, \end{eqnarray*} where $ψ(t)$ is the unique positive solution of the ODE \begin{eqnarray*} ψ' = ψ^p \ln^α(ψ^2 +2), \quad \lim_{t\to T}ψ(t) = + \infty. \end{eqnarray*} The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion. By the interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we show the stability of the constructed solution with respect to perturbations in initial data. To our knowledge, this is the first successful construction for a genuinely non-scale invariant PDE of a stable blowup solution with the derivation of the blowup profile. From this point of view, we consider our result as a breakthrough.

math.AP↗