arXiv subjects

Justin Holmer

Publications and source records attributed to Justin Holmer.

At least 19 recordsLinked to original sources

arXiv2026

Equilibria for the Vlasov-Maxwell system related to plasma confinement

By working in a setting of azimuthal and $z$-directional invariance, we reduce the existence of electrically neutral, magnetically driven, equilibria of the two-species Vlasov-Maxwell (VM) equation to a second-order ODEs in the radial variable. In contrast to previous works, we consider specific situations that correspond physically to $z$-pinch, $\theta$-pinch, and screw-pinch equilibria in plasma physics. Semi-explicit solutions in these situations are provided.

math.AP↗

arXiv2026

Spectral property for the 2D Zakharov-Kuznetsov equation

We discuss a spectral property for the virial operator of the 2D Zakharov-Kuznetsov (ZK) equation. This is a crucial ingredient to establish blow-up or asymptotic stability of solitary waves in higher-dimensional problems. This model in 3D setting was originally introduced by Zakharov and Kuznetsov in plasma physics, and is also a higher-dimensional generalization of the well-known Korteweg-de Vries (KdV) equation. The problem of stability of solitary waves in ZK equation or stable blow-up in modified ZK (or KdV-type) equation is an important physical question, for which virial operators and their spectral properties are the essential elements of the analysis. In this paper we investigate this problem analytically and reduce it to verifying numerically only some signs of inner products and certain eigenvalues.

math.AP↗

arXiv2025

Orbital stability of kinks in the NLS equation with competing nonlinearities

Kinks connecting zero and nonzero equilibria in the NLS equation with competing nonlinearities occur at the special values of the frequency parameter. Since they are minimizers of energy, they are expected to be orbitally stable in the time evolution of the NLS equation. However, the stability proof is complicated by the degeneracy of kinks near the nonzero equilibrium. The main purpose of this work is to give a rigorous proof of the orbital stability of kinks. We give details of analysis for the cubic--quintic NLS equation and show how the proof is extended to the general case.

math.AP↗

arXiv2023

The Derivation of the Boltzmann Equation from Quantum Many-body Dynamics

We consider the quantum many-body dynamics at the weak-coupling scaling. We derive rigorously the quantum Boltzmann equation, which contains the classical hard sphere model and, effectively, the inverse power law model, from the many-body dynamics assuming a physical and optimal regularity bound. The regularity bound we find, on the one hand, is satisfied by quasi-free solutions and comes from calculations regarding the local Maxwellian solution, in which we also prove that 2-body molecular chaos never happens unless $N=+\infty$; on the other hand, it arises from the well-posedness threshold of the limiting Boltzmann equation below which we prove ill-posedness. That is, the regularity cannot be higher at the $N$-body level, cannot be lower in the limit, and is hence a double criticality. To work with this borderline case, we analyze all four sides, with respect to the Fourier transform, of the BBGKY hierarchy sequence with new tools and techniques. We prove well-definedness, compactness, convergence, and uniqueness of hierarchies right at the criticality to complete an optimal derivation. In particular, we have proved that, for physical $N$-particle solutions, the Boltzmann equation emerges as the mean-field limit and time is hence irreversible, from first principles of quantum mechanics.

math-ph↗

arXiv2022

Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel

We consider the 3D Boltzmann equation with the constant collision kernel. We investigate the well/ill-posedness problem using the methods from nonlinear dispersive PDEs. We construct a family of special solutions, which are neither near equilibrium nor self-similar, to the equation, and prove that the well/ill-posedness threshold in $H^{s}$ Sobolev space is exactly at regularity $s=1$, despite the fact that the equation is scale invariant at $s=\frac{1}{2}$.

math.AP↗

arXiv2022

Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space

We consider the derivation of the defocusing cubic nonlinear Schrödinger equation (NLS) on $\mathbb{R}^{3}$ from quantum $N$-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering theorem for the NLS to obtain convergence rate estimates under $H^{1}$ regularity. The $H^{1}$ convergence rate estimate we obtain is almost optimal for $H^{1}$ datum, and immediately improves if we have any extra regularity on the limiting initial one-particle state.

math.AP↗

arXiv2022

Benjamin-Ono Soliton Dynamics in a slowly varying potential revisited

The Benjamin Ono equation with a slowly varying potential is $$ \text{(pBO)} \qquad u_t + (Hu_x-Vu + \tfrac12 u^2)_x=0 $$ with $V(x)=W(hx)$, $0< h \ll 1$, and $W\in C_c^\infty(\mathbb{R})$, and $H$ denotes the Hilbert transform. The soliton profile is $$Q_{a,c}(x) = cQ(c(x-a)) \,, \text{ where } Q(x) = \frac{4}{1+x^2}$$ and $a\in \mathbb{R}$, $c>0$ are parameters. For initial condition $u_0(x)$ to (pBO) close to $Q_{0,1}(x)$, it was shown in a previous work by Z. Zhang that the solution $u(x,t)$ to (pBO) remains close to $Q_{a(t),c(t)}(x)$ and approximate parameter dynamics for $(a,c)$ were provided, on a dynamically relevant time scale. In this paper, we prove exact $(a,c)$ parameter dynamics. This is achieved using the basic framework of the previous work by Z. Zhang but adding a local virial estimate for the linearization of (pBO) around the soliton. This is a local-in-space estimate averaged in time, often called a local smoothing estimate, showing that effectively the remainder function in the perturbation analysis is smaller near the soliton than globally in space. A weaker version of this estimate is proved in a paper by Kenig & Martel as part of a "linear Liouville" result, and we have adapted and extended their proof for our application.

math.AP↗

arXiv2020

The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on $\mathbb{T}^{4}$

We consider the $\mathbb{T}^{4}$ cubic NLS which is energy-critical. We study the unconditional uniqueness of solution to the NLS via the cubic Gross-Pitaevskii hierarchy, an uncommon method, and does not require the existence of solution in Strichartz type spaces. We prove $U$-$V$ multilinear estimates to replace the previously used Sobolev multilinear estimates, which fail on $\mathbb{T}^{4}$. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel-Born expansion. The new combinatorics and the $U$-$V$ estimates then seamlessly conclude the $H^{1}$ unconditional uniqueness for the NLS under the infinite hierarchy framework. This work establishes a unified schemes to prove $H^{1}$ uniqueness for the $\mathbb{R}^{3}/\mathbb{R}^{4}/\mathbb{T}^{3}/\mathbb{T}^{4}$ energy-critical Gross-Pitaevskii hierarchies and thus the corresponding NLS.

math.AP↗

arXiv2020

Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation

We consider the quadratic Zakharov-Kuznetsov equation $$ \partial_t u + \partial_x Δu + \partial_x u^2 =0 $$ on $\mathbb{R}^3$. A solitary wave solution is given by $Q(x-t,y,z)$, where $Q$ is the ground state solution to $-Q + ΔQ + Q^2 =0$. We prove the asymptotic stability of these solitary wave solutions. Specifically, we show that initial data close to $Q$ in the energy space, evolves to a solution that, as $t\to\infty$, converges to a rescaling and shift of $Q(x-t,y,z)$ in $L^2$ in a rightward shifting region $x> δt -\tan θ\sqrt{y^2+z^2} $ for $0 \leq θ\leq \fracπ{3}-δ$.

math.AP↗

arXiv2019

Scattering for the $L^2$ supercritical point NLS

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. "Point" means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of solution, blow-up occurrence and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e, we show that the global solution scatters as t tends to minus/plus infinity in the $L^2$ supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

math.AP↗

arXiv2019

The Derivation of the $\mathbb{T}^{3}$ Energy-critical NLS from Quantum Many-body Dynamics

We derive the 3D energy critical quintic NLS from quantum many-body dynamics with 3-body interaction in the T^3 (periodic) setting. Due to the known complexity of the energy critical setting, previous progress was limited in comparison to the 2-body interaction case yielding energy subcritical cubic NLS. Previously, the only result for the 3D energy critical case was HTX, which proved the uniqueness part of the argument in the case of small solutions. In the main part of this paper, we develop methods to prove the convergence of the BBGKY hierarchy to the infinite Gross-Pitaevskii (GP) hierarchy, and separately, the uniqueness of large GP solutions. Since the trace estimate used in the previous proofs of convergence is the false sharp trace estimate in our setting, we instead introduce a new frequency interaction analysis and apply the finite dimensional quantum de Finetti theorem. For the large solution uniqueness argument, we discover the new HUFL (hierarchical uniform frequency localization) property for the GP hierarchy and use it to prove a new type of uniqueness theorem. The HUFL property reduces to a new statement even for NLS. With the help of CKSTT,IP which proved the global well-posedness for the quintic NLS, this new uniqueness theorem establishes global uniqueness.

math.AP↗

arXiv2018

Blow-up in finite or infinite time of the 2D cubic Zakharov-Kuznetsov equation

We prove that near-threshold negative energy solutions to the 2D cubic ($L^2$-critical) focusing Zakharov-Kuznetsov (ZK) equation blow-up in finite or infinite time. The proof consists of several steps. First, we show that if the blow-up conclusion is false, there are negative energy solutions arbitrarily close to the threshold that are globally bounded in $H^1$ and are spatially localized, uniformly in time. In the second step, we show that such solutions must in fact be exact remodulations of the ground state, and hence, have zero energy, which is a contradiction. This second step, a nonlinear Liouville theorem, is proved by contradiction, with a limiting argument producing a nontrivial solution to a (linear) linearized ZK equation obeying uniform-in-time spatial localization. Such nontrivial linear solutions are excluded by a local-viral space-time estimate. The general framework of the argument is modeled on Merle [29] and Martel & Merle [24], who treated the 1D problem of the $L^2$-critical gKdV equation. Several new features are introduced here to handle the 2D ZK case.

math.AP↗

arXiv2017

Instability of solitons in the 2d cubic Zakharov-Kuznetsov equation

We consider the two dimensional generalization of the Korteweg-de Vries equation, the generalized Zakharov-Kuznetsov (ZK) equation, $u_t + \partial_{x_1}(Δu + u^p) = 0, (x_1,x_2) \in \mathbb{R}^2$. It is known that solitons are stable for nonlinearities $p < 3$ and unstable for $p > 3$, which was established by Anne de Bouard in [5] generalizing the arguments of Bona-Souganidis-Strauss in [1] for the gKdV equation. The $L^2$-critical case with $p=3$ has been open and in this paper we prove that solitons are unstable in the cubic ZK equation. This matches the situation with the critical gKdV equation, proved in 2001 by Martel and Merle in [22]. While the general strategy follows [22], the two dimensional case creates several difficulties and to deal with them, we design a new virial-type quantity, revisit monotonicity properties and, most importantly, develop new pointwise decay estimates, which can be useful in other contexts.

math.AP↗

arXiv2017

Instability of solitons - revisited, I: the critical generalized KdV equation

We revisit the phenomenon of instability of solitons in the generalized Korteweg-de Vries equation, $u_t + \partial_x(u_{xx} + u^p) = 0$. It is known that solitons are unstable for nonlinearities $p \geq 5$, with the critical power $p=5$ being the most challenging case to handle. The critical case was proved by Martel-Merle in [11], where the authors crucially relied on the pointwise decay estimates of the linear KdV flow. In this paper, we show simplified approaches to obtain the instability of solitons via truncation and monotonicity, which can be also useful for other KdV-type equations.

math.AP↗

arXiv2017

Instability of solitons - revisited, II: the supercritical Zakharov-Kuznetsov equation

We revisit the phenomenon of instability of solitons in the two dimensional generalization of the Korteweg-de Vries equation, the generalized Zakharov-Kuznetsov (ZK) equation, $u_t + \partial_{x_1} (Δu + u^p) = 0, (x_1,x_2) \in \mathbb R^2$. It is known that solitons are unstable in this two dimensional equation for nonlinearities $p > 3$. This was shown by Anne de Bouard in [4] generalizing the arguments of Bona-Souganidis-Strauss in [1] for the generalized KdV equation. In this paper, we use a different method to obtain the instability of solitons, namely, truncation and monotonicity properties. Not only does this approach simplify the proof, but it can also be useful for studying various other stability questions in the ZK equation as well as other generalizations of the KdV equation.

math.AP↗

arXiv2017

Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles

We consider the 1D nonlinear Schrödinger equation (NLS) with focusing \emph{point nonlinearity}, $$i\partial_tψ+ \partial_x^2ψ+ δ|ψ|^{p-1}ψ= 0$$ where $δ=δ(x)$ is the delta function supported at the origin. In the $L^2$ supercritical setting $p>3$, we construct self-similar blow-up solutions belonging to the energy space $L_x^\infty \cap \dot H_x^1$. This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic Weber functions and solving the jump condition at $x=0$ imposed by the $δ$ term. This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case $0<p-3 \ll 1$ using the log Binet formula for the gamma function, and contour deformation and stationary phase/Laplace method in the integral formulae for the parabolic Weber functions.

math.AP↗

arXiv2015

Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory

We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity, $$ (δ\text{NLS}) \qquad i\partial_tψ+ \partial_x^2ψ+ δ|ψ|^{p-1}ψ= 0, $$ where $δ=δ(x)$ is the delta function supported at the origin. We show that $δ$NLS shares many properties in common with those previously established for the focusing autonomous translationally-invariant NLS $$ (\text{NLS}) \qquad i\partial_t ψ+ Δψ+ |ψ|^{p-1}ψ=0 \,. $$ The critical Sobolev space $\dot H^{σ_c}$ for $δ$NLS is $σ_c=\frac12-\frac{1}{p-1}$, whereas for NLS it is $σ_c=\frac{d}{2}-\frac{2}{p-1}$. In particular, the $L^2$ critical case for $δ$NLS is $p=3$. We prove several results pertaining to blow-up for $δ$NLS that correspond to key classical results for NLS. Specifically, we (1) obtain a sharp Gagliardo-Nirenberg inequality analogous to Weinstein (1983), (2) apply the sharp Gagliardo-Nirenberg inequality and a local virial identity to obtain a sharp global existence/blow-up threshold analogous to Weinstein (1983), Glassey (1977) in the case $σ_c=0$ and Duyckaerts, Holmer, & Roudenko (2008), Guevara (2014), and Fang, Xie, & Cazenave (2011) for $0<σ_c<1$, (3) prove a sharp mass concentration result in the $L^2$ critical case analogous to Tsutsumi (1990), Merle & Tsutsumi (1990) and (4) show that minimal mass blow-up solutions in the $L^2$ critical case are pseudoconformal transformations of the ground state, analogous to Merle (1993).

math.AP↗

arXiv2015

The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution

We consider a 2D time-dependent quantum system of $N$-bosons with harmonic external confining and \emph{attractive} interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the $k$-th power of the energy controls the $H^{1}$ Sobolev norm of the solution over $k$-particles. This estimate is new and more difficult for attractive interactions than repulsive interactions. For the proof, we use a version of the finite-dimensional quantum di Finetti theorem from [49]. A high particle-number averaging effect is at play in the proof, which is not needed for the corresponding estimate in the repulsive case. This a priori bound allows us to prove that the corresponding BBGKY hierarchy converges to the GP limit as was done in many previous works treating the case of repulsive interactions. As a result, we obtain that the \emph{focusing} nonlinear Schrödinger equation is the mean-field limit of the 2D time-dependent quantum many-body system with attractive interatomic interaction and asymptotically factorized initial data. An assumption on the size of the $L^{1}$-norm of the interatomic interaction potential is needed that corresponds to the sharp constant in the 2D Gagliardo-Nirenberg inequality though the inequality is not directly relevant because we are dealing with a trace instead of a power.

math.AP↗