On the stability of 2D dipolar Bose-Einstein condensates

We study the existence of energy minimizers for a Bose-Einstein condensate with dipole-dipole interactions, tightly confined to a plane. The problem is critical in that the kinetic energy and the (partially attractive) interaction energy behave the same under mass-preserving scalings of the wave-function. We obtain a sharp criterion for the existence of ground states, involving the optimal constant of a certain generalized Gagliardo-Nirenberg inequality

Decay of solitary waves of fractional Korteweg-de Vries type equations

We study the solitary waves of fractional Korteweg-de Vries type equations, that are related to the 1- dimensional semi-linear fractional equations: |D|αu + u − f (u) = 0, with α ∈ (0, 2), a prescribed coefficient p∗(α), and a non-linearity f (u) = |u|p−1 u for p ∈ (1,p∗(α)), or f (u) = up with an integer p ∈ [2;p∗(α)). Asymptotic developments of order 1 at infinity of solutions are given, as well as second order developments for positive solutions, in terms of the coefficient of dispersion α and of the non-linearity p. The main tools are the kernel formulation introduced by Bona and Li, and an accurate description of the kernel by complex analysis theory.publishedVersio

A study of different interactions between solitary waves for fractional Korteweg-de Vries type equations

Denne avhandlingen er satt sammen av tre artikler. I den første konstruerer vi $N$-soliton løsninger for den fractional Korteweg-de Vries (fKdV) ligningen $\partial_t u - \partial_x\left(|D|^{\alpha}u - u^2 \right)=0,$ i hele det underkritiske tilfellet $\alpha \in(\frac12,2)$. Mer presist, hvis $Q_c$ er grunntilstandsløsningen knyttet til fKdV som beveger seg med hastighet $c$, da gitt $0<c_1< \cdots < c_N$, beviser vi eksistensen av en løsning $U$ av (fKdV) som tilfredstiller $\lim_{t\to\infty} \| U(t,\cdot) - \sum_{j=1}^NQ_{c_j}(x-\rho_j(t)) \|_{H^{\frac{\alpha}2}}=0,$ hvor $\rho'_j(t) \sim c_j$ som $t \to +\infty$. Beviset er basert på konstruksjonen gjort av Martel for den generaliserte KdV-ligningen [Amer. J. Math. 127 (2005), s. 1103-1140]) for ikke-lokale ligninger. De største utfordringene i dette arbeidet er knyttet til egenskapene av grunntilstanden $Q_c$. Mer presist, så avtar funksjonen som et algebraisk polynom. Samt, er det utfordringer knyttet til bruken av lokale teknikker (monotomiegenskaper for en del av massen og energien) for en ikke-lokal ligning. For å omgå disse vanskelighetene bruker vi symmetriske og ikke-symmetriske vektede kommutatorestimater. De symmetriske estimatene ble bevist av Kenig, Martel og Robbiano [Annales de l'IHP Analyze Non Linéaire 28 (2011), s. 853-887], mens de ikke-symmetriske estimatene ser ut til å være nye. I den andre artikkelen studerer vi den fraksjonale ikke-lineære Schrödinger-ligningen i dimensjon en: $\vert D \vert^\alpha u + u -f(u)=0,$ med $\alpha\in (0,2)$, en gitt koeffisient $p^*(\alpha)$, og en ikke-linæritet $f(u)=\vert u \vert^{p-1}u$ for $p\in(1,p^*(\alpha))$, eller $f(u)=u^p$ med et heltall $p\in[2;p^*(\alpha))$. Vi gir asymptotiske utviklinger av løsningen til første orden ved uendelig. Samt, gir vi andreordens utviklinger for positive løsninger. Disse asymptotiske utviklingene er avhenger av dispersjonskoeffisienten $\alpha$ og ikke-linæriteten $p$. Hovedverktøyene er kernelformuleringen introdusert av Bona og Li [J. Math. Pures Appl. (9) 76 (1997), no. 5, 377-430], og en nøyaktig beskrivelse av kernelen ved hjelp av kompleks analyse. I den siste artikkelen studerer vi en spesiell asymptotisk oppførsel av en dipolløsning av den fractional modifiserte Korteweg-de Vries-ligningen: $\partial_t u + \partial_x (-\vert D \vert^\alpha u + u^3)=0.$ Dipolløsningen er en løsning som oppfører seg som en sum av to sterkt interaktive solitære bølger med forskjellige fortegn, når tiden er stor nok. Vi beviser eksistensen av en dipol for fmKdV. Et viktig bidrag i denne artikkelen er konstruksjonen av nøyaktige profiler, og dette er nytt for fmKdV ligningen. Dessuten, for å håndtere den ikke-lokale operatoren $\vert D \vert^\alpha$, må vi utbedre noen vektede kommutatorestimater.This thesis in composed by three articles. In the first one, we construct $N$-soliton solutions for the fractional Korteweg-de Vries (fKdV) equation $\partial_t u - \partial_x\left(|D|^{\alpha}u - u^2 \right)=0,$ in the whole sub-critical range $\alpha \in(\frac12,2)$. More precisely, if $Q_c$ denotes the ground state solution associated with fKdV evolving with velocity $c$, then given $0<c_1< \cdots < c_N$, we prove the existence of a solution $U$ of (fKdV) satisfying $\lim_{t\to\infty} \| U(t,\cdot) - \sum_{j=1}^NQ_{c_j}(x-\rho_j(t)) \|_{H^{\frac{\alpha}2}}=0,$ where $\rho'_j(t) \sim c_j$ as $t \to +\infty$. The proof adapts the construction of Martel in the generalized KdV setting [Amer. J. Math. 127 (2005), pp. 1103-1140]) to the fractional case. The main new difficulties are the polynomial decay of the ground state $Q_c$ and the use of local techniques (monotonicity properties for a portion of the mass and the energy) for a non-local equation. To bypass these difficulties, we use symmetric and non-symmetric weighted commutator estimates. The symmetric ones were proved by Kenig, Martel and Robbiano [Annales de l'IHP Analyse Non Linéaire 28 (2011), pp. 853-887], while the non-symmetric ones seem to be new. In the second paper, we consider the fractional nonlinear Schrödinger equation in dimension $1$: $\vert D \vert^\alpha u + u -f(u)=0,$ with $\alpha\in (0,2)$, a prescribed coefficient $p^*(\alpha)$, and a non-linearity $f(u)=\vert u \vert^{p-1}u$ for $p\in(1,p^*(\alpha))$, or $f(u)=u^p$ with an integer $p\in[2;p^*(\alpha))$. Asymptotic developments of order $1$ of the solutions at infinity are given, as well as second order developments for positive solutions, in terms of the coefficient of dispersion $\alpha$ and of the non-linearity $p$. The main tools are the kernel formulation introduced by Bona and Li [J. Math. Pures Appl. (9) 76 (1997), no. 5, 377-430], and an accurate description of the kernel by complex analysis theory. In the last paper, we study one particular asymptotic behaviour of a solution of the fractional modified Korteweg-de Vries equation (also known as the dispersion generalised modified Benjamin-Ono equation): $\partial_t u + \partial_x (-\vert D \vert^\alpha u + u^3)=0.$ The dipole solution is a solution behaving in large time as a sum of two strongly interacting solitary waves with different signs. We prove the existence of a dipole for fmKdV. A novelty of this article is the construction of accurate profiles. Moreover, to deal with the non-local operator $\vert D \vert^\alpha$, we refine some weighted commutator estimates.Doktorgradsavhandlin

STRONGLY INTERACTING SOLITARY WAVES FOR THE FRACTIONAL MODIFIED KORTEWEG-DE VRIES EQUATION

We study one particular asymptotic behaviour of a solution of the fractional modified Korteweg-de Vries equation (also known as the dispersion generalised modified Benjamin-Ono equation): ∂tu + ∂x(−|D| α u + u 3) = 0. (fmKdV) The dipole solution is a solution behaving in large time as a sum of two strongly interacting solitary waves with different signs. We prove the existence of a dipole for fmKdV. A novelty of this article is the construction of accurate profiles. Moreover, to deal with the non-local operator |D| α , we refine some weighted commutator estimates

On the stability of 2D dipolar Bose-Einstein condensates

International audienceWe study the existence of energy minimizers for a Bose-Einstein condensate with dipole-dipole interactions, tightly confined to a plane. The problem is critical in that the kinetic energy and the (partially attractive) interaction energy behave the same under mass-preserving scalings of the wave-function. We obtain a sharp criterion for the existence of ground states, involving the optimal constant of a certain generalized Gagliardo-Nirenberg inequality

Low efficiency of large volcanic eruptions in transporting very fine ash into the atmosphere

Abstract Volcanic ash clouds are common, often unpredictable, phenomena generated during explosive eruptions. Mainly composed of very fine ash particles, they can be transported in the atmosphere at great distances from the source, having detrimental socio-economic impacts. However, proximal settling processes controlling the proportion (ε) of the very fine ash fraction distally transported in the atmosphere are still poorly understood. Yet, for the past two decades, some operational meteorological agencies have used a default value of ε = 5% as input for forecast models of atmospheric ash cloud concentration. Here we show from combined satellite and field data of sustained eruptions that ε actually varies by two orders of magnitude with respect to the mass eruption rate. Unexpectedly, we demonstrate that the most intense eruptions are in fact the least efficient (with ε = 0.1%) in transporting very fine ash through the atmosphere. This implies that the amount of very fine ash distally transported in the atmosphere is up to 50 times lower than previously anticipated. We explain this finding by the efficiency of collective particle settling in ash-rich clouds which enhance early and en masse fallout of very fine ash. This suggests that proximal sedimentation during powerful eruptions is more controlled by the concentration of ash than by the grain size. This has major consequences for decision-makers in charge of air traffic safety regulation, as well as for the understanding of proximal settling processes. Finally, we propose a new statistical model for predicting the source mass eruption rate with an unprecedentedly low level of uncertainty

Author Correction: Low efficiency of large volcanic eruptions in transporting very fine ash into the atmosphere

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Eruption type probability and eruption source parameters at Cotopaxi and Guagua Pichincha volcanoes (Ecuador) with uncertainty quantification

Co-auteur étrangerInternational audienceFuture occurrence of explosive eruptive activity at Cotopaxi and Guagua Pichincha volcanoes, Ecuador, is assessed probabilistically, utilizing expert elicitation. Eight eruption types were considered for each volcano. Type event probabilities were evaluated for the next eruption at each volcano and for at least one of each type within the next 100 years. For each type, we elicited relevant eruption source parameters (duration, average plume height, and total tephra mass). We investigated the robustness of these elicited evaluations by deriving probability uncertainties using three expert scoring methods. For Cotopaxi, we considered both rhyolitic and andesitic magmas. Elicitation findings indicate that the most probable next eruption type is an andesitic hydrovolcanic/ash-emission (~ 26–44% median probability), which has also the highest median probability of recurring over the next 100 years. However, for the next eruption at Cotopaxi, the average joint probabilities for sub-Plinian or Plinian type eruption is of order 30–40%—a significant chance of a violent explosive event. It is inferred that any Cotopaxi rhyolitic eruption could involve a longer duration and greater erupted mass than an andesitic event, likely producing a prolonged emergency. For Guagua Pichincha, future eruption types are expected to be andesitic/dacitic, and a vulcanian event is judged most probable for the next eruption (median probability ~40–55%); this type is expected to be most frequent over the next 100 years, too. However, there is a substantial probability (possibly >40% in average) that the next eruption could be sub-Plinian or Plinian, with all that implies for hazard levels