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Mihaela Ifrim

Publications and source records attributed to Mihaela Ifrim.

At least 37 records · Page 2Linked to original sources
arXiv2022

Testing by wave packets and modified scattering in nonlinear dispersive pde's

Modified scattering phenomena are encountered in the study of global properties for nonlinear dispersive partial differential equations in situations where the decay of solutions at infinity is borderline and scattering fails just barely. An interesting example is that of problems with cubic nonlinearities in one space dimension. The method of testing by wave packets was introduced by the authors as a tool to efficiently capture the asymptotic equations associated to such flows, and thus establish the modified scattering mechanism in a simpler, more efficient fashion, and at lower regularity. In these expository notes we describe how this method can be applied to problems with general dispersion relations.

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arXiv2022

Long time solutions for 1D cubic dispersive equations, Part II: the focusing case

This article is concerned with one dimensional dispersive flows with cubic nonlinearities on the real line. In a very recent work, the authors have introduced a broad conjecture for such flows, asserting that in the defocusing case, small initial data yields global, scattering solutions. Then this conjecture was proved in the case of a Schrödinger dispersion relation. In terms of scattering, our global solutions were proved to satisfy both global $L^6$ Strichartz estimates and bilinear $L^2$ bounds. Notably, no localization assumption is made on the initial data. In this article we consider the focusing scenario. There potentially one may have small solitons, so one cannot hope to have global scattering solutions in general. Instead, we look for long time solutions, and ask what is the time-scale on which the solutions exist and satisfy good dispersive estimates. Our main result, which also applies in the case of the Schrödinger dispersion relation, asserts that for initial data of size $ε$, the solutions exist on the time-scale $ε^{-8}$, and satisfy the desired $L^6$ Strichartz estimates and bilinear $L^2$ bounds on the time-scale $ε^{-6}$. To the best of our knowledge, this is the first result to reach such a threshold.

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arXiv2022

The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity

This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.

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arXiv2022

Local well-posedness for quasilinear problems: a primer

Proving local well-posedness for quasilinear problems in pde's presents a number of difficulties, some of which are universal and others of which are more problem specific. While a common standard, going back to Hadamard, has existed for a long time, there are by now both many variations and many misconceptions in the subject. The aim of these notes is to collect a number of both classical and more recent ideas in this direction, and to assemble them into a cohesive road map that can be then adapted to the reader's problem of choice.

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arXiv2022

The relativistic Euler equations with a physical vacuum boundary: Hadamard local well-posedness, rough solutions, and continuation criterion

In this paper we provide a complete local well-posedness theory for the free boundary relativistic Euler equations with a physical vacuum boundary on a Minkowski background. Specifically, we establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the the velocity is in $L^1_t Lip$ and a suitable weighted version of the density is at the same regularity level. Our entire approach is in Eulerian coordinates and relies on the functional framework developed in the companion work of the second and third authors corresponding to the non relativistic problem. All our results are valid for a general equation of state $p(\varrho)= \varrho^γ$, $γ> 1$.

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arXiv2022

Dispersive decay of small data solutions for the KdV equation

We consider the Korteweg-de Vries (KdV) equation, and prove that small localized data yields solutions which have dispersive decay on a quartic time-scale. This result is optimal, in view of the emergence of solitons at quartic time, as predicted by inverse scattering theory.

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arXiv2021

The time-like minimal surface equation in Minkowski space: low regularity solutions

It has long been conjectured that for nonlinear wave equations which satisfy a nonlinear form of the null condition, the low regularity well-posedness theory can be significantly improved compared to the sharp results of Smith-Tataru for the generic case. The aim of this article is to prove the first result in this direction, namely for the time-like minimal surface equation in the Minkowski space-time. Further, our improvement is substantial, namely by $3/8$ derivatives in two space dimensions and by $1/4$ derivatives in higher dimensions.

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arXiv2021

Two dimensional gravity waves at low regularity II: Global solutions

This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. Such solutions have been proved to exist earlier in [15, 7, 10, 12] in much higher regularity. Our goal in this paper is to improve these results and prove global well-posedness under minimal regularity and decay assumptions for the initial data. One key ingredient here is represented by the balanced cubic estimates in our first paper. Another is the nonlinear vector field Sobolev inequalities, an idea first introduced by the last two authors in the context of the Benjamin-Ono equations [14].

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arXiv2021

No pure capillary solitary waves exist in 2D finite depth

We prove that the 2D finite depth capillary water wave equations admit no solitary wave solutions. This closes the existence/non-existence problem for solitary water waves in 2D, under the classical assumptions of incompressibility and irrotationality, and with the physical parameters being gravity, surface tension and the fluid depth.

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arXiv2019

A Morawetz inequality for gravity-capillary water waves at low Bond number

This paper is devoted to the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the general question of proving Morawetz inequalities. We continue the analysis initiated in our previous work, where we have established local energy decay estimates for gravity waves. Here we add surface tension and prove a stronger estimate with a local regularity gain, akin to the smoothing effect for dispersive equations. Our main result holds globally in time and holds for genuinely nonlinear waves, since we are only assuming some very mild uniform Sobolev bounds for the solutions. Furthermore, it is uniform both in the infinite depth limit and the zero surface tension limit.

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arXiv2019

No solitary waves in 2-d gravity and capillary waves in deep water

A fundamental question in the study of water waves is the existence and stability of solitary waves. Solitary waves have been proved to exist and have been studied in many interesting situations, and often arise from the balance of different forces/factors influencing the fluid dynamics, e.g. gravity, surface tension or the fluid bottom. However, the existence of solitary waves has remained an open problem in two of the simplest cases, namely for either pure gravity waves or pure capillary waves in infinite depth. In this article we settle both of these questions in two space dimensions. Precisely, we consider incompressible, irrotational, infinite depth water wave equation, either with gravity and without surface tension, or without gravity but with surface tension. In both of these cases we prove that there are no solitary waves.

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arXiv2018

The NLS approximation for two dimensional deep gravity waves

This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schrödinger equation (NLS) on the natural cubic time scale.

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arXiv2018

A Morawetz inequality for water waves

We consider gravity water waves in two space dimensions, with finite or infinite depth. Assuming some uniform scale invariant Sobolev bounds for the solutions, we prove local energy decay (Morawetz) estimates globally in time. Our result is uniform in the infinite depth limit.

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arXiv2017

The lifespan of small data solutions in two dimensional capillary water waves

This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions.

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arXiv2017

Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation

This article represents a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. Here, we prove that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach.

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arXiv2017

Two dimensional gravity water waves with constant vorticity: I. Cubic lifespan

This article is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small data solutions.

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arXiv2016

Finite depth gravity water waves in holomorphic coordinates

In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match the earlier infinite depth result of Hunter-Ifrim-Tataru.

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