arXiv subjects

Van Tien Nguyen

Publications and source records attributed to Van Tien Nguyen.

At least 19 recordsLinked to original sources

arXiv2026

Axisymmetric type II blowup solutions to the three-dimensional Keller-Segel system

We construct axisymmetric solutions to the three-dimensional parabolic-elliptic Keller-Segel system that blow up in finite time. In particular, the singularity is of type II, which locally admits a leading-order profile of the rescaled stationary solution of the two-dimensional system. Additionally, mass concentration occurs along a one-dimensional ring in the plane. In the analysis, we rely on an approximate solution of the eigenproblem associated with the linearized operator around the stationary solution as well as the modulation dynamics to control the perturbation function and derive the accurate blowup rate.

math.AP↗

arXiv2026

Singularity formed by the collision of two collapsing solitons in interaction for the 2D Keller-Segel system

It is well-known that the two-dimensional parabolic-elliptic Keller-Segel system admits finite-time blowup solutions, which is the case if the initial density has total mass greater than $8π$. Several constructive examples of such solutions have been given, where for all of them a perturbed stationary state undergoes scale instability and collapses at a point, resulting in an $8π$-mass concentration. It was conjectured that singular solutions concentrating more than one soliton simultaneously could exist. We construct rigorously such a new blowup mechanism, where two stationary states are simultaneously collapsing and colliding, resulting in a $16π$-mass concentration at a single blowup point, and with a new blowup rate which corresponds to the formal prediction by Seki, Sugiyama and Velázquez. We develop, for the first time, a robust framework to rigorously construct blowup solutions that simultaneously involve the non-radial collision and concentration of several solitons, which we expect to have applications to other evolution problems.

math.AP↗

arXiv2026

$L^2$-based stability of blowup with log correction for semilinear heat equation

We propose an alternative proof of the classical result of Type-I blowup with log correction for the semilinear heat equation. Compared with previous proofs, we use a novel idea of enforcing stable normalizations for perturbations around the approximate profile and we establish a weighted $H^k$ stability, thereby avoiding the use of a topological argument and the analysis of a linearized spectrum. Consequently, this approach can be adopted even if we only have a numerical profile and do not have explicit information on the spectrum of its linearized operator. This result generalizes the $L^2$-based stability framework beyond exactly self-similar blowup and can be adapted to higher dimensions. Numerical results corroborate the effectiveness of our normalization, even in the large perturbation regime beyond our theoretical setting.

math.AP↗

arXiv2025

Nonradial Quenching Profile for a MEMS Model

We construct a quenching solution to the parabolic MEMS model \[ u_t = Δu - \frac{1}{u^2} \quad \text{in } \mathcal{B} \times (0,T), \quad u|_{\partial \mathcal{B}} = 1, \] where $\mathcal{B}$ is the unit disc in $\mathbb{R}^2$, and $T > 0$ denotes the quenching time. The constructed solution quenches only at the origin and admits the final profile \[ u(x,T) \sim \left(x_1^2 x_2^2 + θ(x_1^6 + x_2^6)\right)^{\frac{1}{3}} \quad \text{as } |x| \to 0, \] where $θ\in (0, θ^*)$ for some $θ^* > 0$. To our knowledge, this is the first example of a quenching solution with a genuinely non-radial profile. The proof relies on the construction of a good approximate solution, using a perturbative expansion in self-similar variables. We then justify the true solution that remains close to this approximation through a spectral analysis combined with a robust energy method.

math.AP↗

arXiv2025

Infinitely many self-similar blow-up profiles for the Keller-Segel system in dimensions 3 to 9

Based on the method of matched asymptotic expansions and Banach fixed point theorem, we rigorously construct infinitely many self-similar blow-up profiles for the parabolic-elliptic Keller-Segel system \begin{equation*} \left\{\begin{array}{l} \partial_{t} u=Δu-\nabla \cdot\left(u \nabla Φ_{u}\right), \\ 0=ΔΦ_{u}+u,\\ u(\cdot,0)=u_0 \geq 0 \end{array}\quad \text{in}\ \mathbb{R}^{d},\right. \end{equation*} where $d\in \{3,\cdots,9\}$. Our findings demonstrate that the infinitely many backward self-similar profiles approximate the rescaling radial steady-state near the origin (i.e. $0<|x|\ll1$) and $\frac{2(d-2)}{|x|^2}$ at spatial infinity (i.e. $|x|\gg1$). We also establish the convergence of the self-similar blow-up solutions as time tends to the blow-up time $T>0$. Our results can give a refined description of backward self-similar profiles for all $|x|\geq 0$ rather than for $0<|x|\ll1$ or $|x|\gg1$, indicating that the blow-up point is the origin and $$ u(x,t)\sim \frac{1}{|x|^2},\ \ \ x\ne0,\ \text{as}\ t\to T. $$

math.AP↗

arXiv2024

On the stability of blowup solutions to the complex Ginzburg-Landau equation in R^d

Building upon the idea in \cite{HNWarXiv24}, we establish stability of the type-I blowup with log correction for the complex Ginzburg-Landau equation. In the amplitude-phase representation, a generalized dynamic rescaling formulation is introduced, with modulation parameters capturing the spatial translation and rotation symmetries of the equation and novel additional modulation parameters perturbing the scaling symmetry. This new formulation provides enough degrees of freedom to impose normalization conditions on the rescaled solution, completely eliminating the unstable and neutrally stable modes of the linearized operator around the blowup profile. It enables us to establish the full stability of the blowup by enforcing vanishing conditions via the choice of normalization and using weighted energy estimates, without relying on a topological argument or a spectrum analysis. The log correction for the blowup rate is captured by the energy estimates and refined estimates of the modulation parameters.

math.AP↗

arXiv2022

Industry-Scale Orchestrated Federated Learning for Drug Discovery

To apply federated learning to drug discovery we developed a novel platform in the context of European Innovative Medicines Initiative (IMI) project MELLODDY (grant n°831472), which was comprised of 10 pharmaceutical companies, academic research labs, large industrial companies and startups. The MELLODDY platform was the first industry-scale platform to enable the creation of a global federated model for drug discovery without sharing the confidential data sets of the individual partners. The federated model was trained on the platform by aggregating the gradients of all contributing partners in a cryptographic, secure way following each training iteration. The platform was deployed on an Amazon Web Services (AWS) multi-account architecture running Kubernetes clusters in private subnets. Organisationally, the roles of the different partners were codified as different rights and permissions on the platform and administrated in a decentralized way. The MELLODDY platform generated new scientific discoveries which are described in a companion paper.

cs.LG↗

arXiv2022

Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher

We consider the parabolic-elliptic Keller-Segel system in three dimensions and higher, corresponding to the mass supercritical case. We construct rigorously a solution which blows up in finite time by having its mass concentrating near a ring that shrinks to a point. In particular, the singularity is of type II, non self-similar. We show the stability of this dynamics among spherically symmetric solutions. In renormalised variables, the solution ressembles a traveling wave imploding at the origin, and this, to our knowledge, is the first stability result for such phenomenon for an evolution PDE. We develop a framework to handle the interactions between the two blowup zones contributing to the mechanism: a thin inner zone around the ring where viscosity effects occur, and an outer zone where the evolution is mostly inviscid.

math.AP↗

arXiv2021

Refined description and stability for singular solutions of the 2D Keller-Segel system

We construct solutions to the two dimensional parabolic-elliptic Keller-Segel model for chemotaxis that blow up in finite time $T$. The solution is decomposed as the sum of a stationary state concentrated at scale $λ$ and of a perturbation. We rely on a detailed spectral analysis for the linearized dynamics in the parabolic neighbourhood of the singularity performed by the authors, providing a refined expansion of the perturbation. Our main result is the construction of a stable dynamics in the full nonradial setting for which the stationary state collapses with the universal law $λ\sim 2e^{-\frac{2+γ}{2}}\sqrt{T-t}e^{-\sqrt{\frac{|\ln (T-t)|}{2}}}$ where $γ$ is the Euler constant. This improves on the earlier result by Raphael and Schweyer 2014 and gives a new robust approach to so-called type II singularities for critical parabolic problems. A by-product of the spectral analysis we developed is the existence of unstable blowup dynamics with speed $λ_\ell \sim C_0(T-t)^{\frac{\ell}{2}} |\ln(T-t)|^{-\frac{\ell}{2(\ell - 1)}}$ for $\ell \geq 2$ integer.

math.AP↗

arXiv2020

Spectral analysis for singularity formation of the two dimensional Keller-Segel system

We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state. This is a two-scale problem, with a vanishing thin transition zone near the origin. Via rigorous matched asymptotic expansions, we describe the eigenvalues and eigenfunctions precisely. We also show a stability result with respect to suitable perturbations, as well as a coercivity estimate for the non-radial part. These results are used as key arguments in a new rigorous proof of the existence and refined description of singular solutions for the Keller-Segel problem by the authors. The present paper extends the result by Dejak, Lushnikov, Yu, Ovchinnikov and Sigal [Physica D, 2012]. Two major difficulties arise in the analysis: this is a singular limit problem, and a degeneracy causes corrections not being polynomial but logarithmic with respect to the main parameter.

math.AP↗

arXiv2018

Construction and Stability of type I blowup solutions for non-variational semilinear parabolic systems

We consider in this note the semilinear heat system $$\partial_t u = Δu + f(v), \quad \partial_t v = μΔv + g(u), \quad μ> 0,$$ where the nonlinearity has no gradient structure taking of the particular form $$f(v) = v|v|^{p-1} \quad \text{and}\quad g(u) = u|u|^{q-1} \quad \text{with} \quad p, q > 1, $$ or $$f(v) = e^{pv}\quad \text{and} \quad g(u) = e^{qu} \quad \text{with} \quad p,q > 0.$$ We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on two-step procedure: the reduction of the problem to a finite dimensional one via a spectral analysis, then solving the finite dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [Ann. IHP 2018] and [JDEs 2018].

math.AP↗

arXiv2018

Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

We consider the energy supercritical wave maps from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation $$\partial_t^2 u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d-1)}{2r^2}\sin(2u).$$ We construct for this equation a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of the universal profile $$u(r,t) \sim Q\left(\frac{r}{λ(t)}\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the quantized rates $$λ(t) \sim c_u(T-t)^\frac{\ell}γ, \quad \ell \in \mathbb{N}^*, \;\; \ell > γ= γ(d) \in (1,2].$$ The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski for the energy supercritical nonlinear Schrödinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.

math.AP↗

arXiv2018

Construction of type I blowup solutions for a higher order semilinear parabolic equation

We consider the higher-order semilinear parabolic equation $$ \partial_t u = -(-Δ)^{m} u + u|u|^{p-1}, $$ in the whole space $\mathbb{R}^N$, where $p > 1$ and $m \geq 1$ is an odd integer. We exhibit type I non self-similar blowup solutions for this equation and obtain a sharp description of its asymptotic behavior. The method of construction relies on the spectral analysis of a non self-adjoint linearized operator in an appropriate scaled variables setting. In view of known spectral and sectorial properties of the linearized operator obtained by [Galaktionov, rspa2011], we revisit the technique developed by [Merle-Zaag, duke1997] for the classical case $m = 1$, which consists in two steps: the reduction of the problem to a finite dimensional one, then solving the finite dimensional problem by a classical topological argument based on the index theory. Our analysis provides a rigorous justification of a formal result in [Galaktionov, rspa2011].

math.AP↗

arXiv2018

Blowup solutions for a reaction-diffusion system with exponential nonlinearities

We consider the following parabolic system whose nonlinearity has no gradient structure: $$\left\{\begin{array}{ll} \partial_t u = Δu + e^{pv}, \quad & \partial_t v = μΔv + e^{qu}, u(\cdot, 0) = u_0, \quad & v(\cdot, 0) = v_0, \end{array}\right. \quad p, q, μ> 0, $$ in the whole space $\mathbb{R}^N$. We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem.

math.AP↗

arXiv2017

On the stability of type II blowup for the 1-corotational energy supercritical harmonic heat flow

We consider the energy supercritical harmonic heat flow from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear heat equation $$\partial_t u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d-1)}{2r^2}\sin(2u).$$ We construct for this equation a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of the universal profile $$u(r,t) \sim Q\left(\frac{r}{λ(t)}\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the quantized rates $$λ(t) \sim c_u(T-t)^\frac{\ell}γ, \quad \ell \in \mathbb{N}^*, \;\; 2\ell > γ= γ(d) \in (1,2].$$ The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski [Camb. Jour. Math, 3(4):439-617, 2015] for the energy supercritical nonlinear Schrödinger equation and by Raphaël and Schweyer [Anal. PDE, 7(8):1713-1805, 2014] for the energy critical harmonic heat flow, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem. Moreover, our constructed solutions are in fact $(\ell - 1)$ codimension stable under perturbations of the initial data. As a consequence, the case $\ell = 1$ corresponds to a stable type II blowup regime.

math.AP↗

arXiv2016

Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term: $$\partial_t U = ΔU + α|\nabla U|^2 + e^U,\quad (x, t)\in\mathbb{R}^N\times[0,T), \quad α> -1.$$ We construct for this equation a solution which blows up in finite time $T > 0$ and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time $T$ simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some senses, resulting in the change of the final blowup profile in comparison with the case $α= 0$. The proof of the construction inspired by the method of Merle and Zaag in 1997, relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. One of the major difficulties arising in the proof is that outside the \textit{blowup region}, the spectrum of the linearized operator around the profile can never be made negative. Truly new ideas are needed to achieve the control of the outer part of the solution. Thanks to a geometrical interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we obtain the stability of the constructed solution with respect to perturbations of the initial data.

math.AP↗

arXiv2016

Refined regularity of the blow-up set linked to refined asymptotic behavior for the semilinear heat equation

We consider $u(x,t)$, a solution of $\partial_tu = Δu + |u|^{p-1}u$ which blows up at some time $T > 0$, where $u:\mathbb{R}^N \times[0,T) \to \mathbb{R}$, $p > 1$ and $(N-2)p < N+2$. Define $S \subset \mathbb{R}^N$ to be the blow-up set of $u$, that is the set of all blow-up points. Under suitable nondegeneracy conditions, we show that if $S$ contains a $(N-\ell)$-dimensional continuum for some $\ell \in \{1,\dots, N-1\}$, then $S$ is in fact a $\mathcal{C}^2$ manifold. The crucial step is to derive a refined asymptotic behavior of $u$ near blow-up. In order to obtain such a refined behavior, we have to abandon the explicit profile function as a first order approximation and take a non-explicit function as a first order description of the singular behavior. This way we escape logarithmic scales of the variable $(T-t)$ and reach significant small terms in the polynomial order $(T-t)^μ$ for some $μ> 0$. The refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on $S$.

math.AP↗

arXiv2016

Construction and stability of blowup solutions for a non-variational semilinear parabolic system

We consider the following parabolic system whose nonlinearity has no gradient structure: $$\left\{\begin{array}{ll} \partial_t u = Δu + |v|^{p-1}v, \quad & \partial_t v = μΔv + |u|^{q - 1}u,\\ u(\cdot, 0) = u_0, \quad & v(\cdot, 0) = v_0, \end{array}\right. $$ in the whole space $\mathbb{R}^N$, where $p, q > 1$ and $μ> 0$. We show the existence of initial data such that the corresponding solution to this system blows up in finite time $T(u_0, v_0)$ simultaneously in $u$ and $v$ only at one blowup point $a$, according to the following asymptotic dynamics: $$\left\{\begin{array}{c} u(x,t)\sim Γ\left[(T-t) \left(1 + \dfrac{b|x-a|^2}{(T-t)|\log (T-t)|}\right)\right]^{-\frac{(p + 1)}{pq - 1}},\\ v(x,t)\sim γ\left[(T-t) \left(1 + \dfrac{b|x-a|^2}{(T-t)|\log (T-t)|}\right)\right]^{-\frac{(q + 1)}{pq - 1}}, \end{array}\right.$$ with $b = b(p,q,μ) > 0$ and $(Γ, γ) = (Γ(p,q), γ(p,q))$. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case $μ= 1$; and the fact that the case $μ\ne 1$ breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.

math.AP↗