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Alexandru D. Ionescu

Publications and source records attributed to Alexandru D. Ionescu.

At least 37 records · Page 2Linked to original sources
arXiv2017

Global regularity for 2d water waves with surface tension

We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of our analysis is to develop a sufficiently robust method, based on energy estimates and dispersive analysis, which allows us to deal with strong singularities arising from time resonances in the applications of the normal form method and nonlinear scattering. As a result, we are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions. Part of our analysis relies on a new treatment of the Dirichlet-Neumann operator in dimension two which is of independent interest. As a consequence, the results in this paper are self-contained.

math.AP↗
arXiv2016

The Euler--Maxwell system for electrons: global solutions in $2D$

A basic model for describing plasma dynamics is given by the Euler-Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. In this paper we consider the "one-fluid" Euler--Maxwell model for electrons, in 2 spatial dimensions, and prove global stability of a constant neutral background.

math.AP↗
arXiv2014

Global solutions for the gravity water waves system in 2d

We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu \cite{WuAG}. We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case \cite{GMS2,Wu3DWW}. In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2d.

math.AP↗
arXiv2013

On the absence of "splash" singularities in the case of two-fluid interfaces

We show that "splash" singularities cannot develop in the case of locally smooth solutions of the two-fluid interface in two dimensions. More precisely, we show that the scenario of formation of singularities discovered by Castro-Córdoba-Fefferman-Gancedo-Gómez-Serrano in the case of the water waves system, in which the interface remains locally smooth but self-intersects in finite time, is completely prevented in the case of two-fluid interfaces with positive densities.

math.AP↗
arXiv2013

The Euler-Maxwell two-fluid system in 3D

The fundamental "two-fluid" model for describing plasma dynamics is given by the Euler-Maxwell system, in which compressible ion and electron fluids interact with their own self-consistent electromagnetic field. We prove global stability of a constant neutral background, in the sense that irrotational, smooth and localized perturbations of a constant background with small amplitude lead to global smooth solutions in three space dimensions for the Euler-Maxwell system. Our construction applies equally well to other plasma models such as the Euler-Poisson system for two-fluids and a relativistic Euler-Maxwell system for two fluids. Our solutions appear to be the first nontrivial global smooth solutions in all of these models.

math.AP↗
arXiv2012

Nonlinear fractional Schrödinger equations in one dimension

We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\partial_t u - Λu = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3, \qquad Λ= Λ(\partial_x) = {|\partial_x|}^(1/2)$$, where $c_0\in\mathbb{R}$ and $c_1,c_2,c_3\in\mathbb{C}$. This model is motivated by the two-dimensional water waves equations, which have a somewhat similar structure in the Eulerian formulation, in the case of irrotational flows. We show that one cannot expect linear scattering, even in this simplified model. More precisely, we identify a suitable nonlinear logarithmic correction, and prove global existence and modified scattering of solutions.

math.AP↗
arXiv2010

On the global well-posedness of energy-critical Schrödinger equations in curved spaces

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global Morawetz inequality adapted to the geometry of the manifold (in other words we adapt the method of Kenig-Merle to the variable coefficient case), and a good understanding of the corresponding Euclidean problem (in our case the main theorem of Colliander-Keel-Staffilani-Takaoka-Tao). As an application we prove global well-posedness and scattering in $H^1$ for the energy-critical defocusing initial-value problem (i\partial_t+Δ_\g)u=u|u|^{4} on the hyperbolic space $H^3$.

math.AP↗
arXiv2009

A para-differential renormalization technique for nonlinear dispersive equations

For α\in (1,2) we prove that the initial-value problem \partial_t u+D^α\partial_x u+\partial_x(u^2/2)=0 on \mathbb{R}_x\times\mathbb{R}_t; u(0)=ϕ, is globally well-posed in the space of real-valued L^2-functions. We use a frequency dependent renormalization method to control the strong low-high frequency interactions.

math.AP↗
arXiv2008

Uniqueness results for ill posed characteristic problems in curved space-times

We prove two uniqueness theorems for solutions of linear and nonlinear wave equations; the first theorem is in the Minkowski space while the second is in the domain of outer communication of a Kerr black hole. Both theorems concern ill posed Cauchy problems on smooth, bifurcate, characteristic hypersurfaces. In the case of the Kerr space-time this hypersurface is the event horizon of the black hole.

gr-qc↗
arXiv2008

On the uniqueness of smooth, stationary black holes in vacuum

We prove a conditional "no hair" theorem for smooth manifolds: if $E$ is the domain of outer communication of a smooth, regular, stationary Einstein vacuum, and if a technical condition relating the Ernst potential and Killing scalar is satisfied on the bifurcate sphere, then $E$ is locally isometric to the domain of outer communication of a Kerr space-time.

gr-qc↗
arXiv2008

Global Schrödinger maps

We consider the Schrödinger map initial-value problem in dimension two or greater. We prove that the Schrödinger map initial-value problem admits a unique global smooth solution, provided that the initial data is smooth and small in the critical Sobolev space. We prove also that the solution operator extends continuously to the critical Sobolev space.

math.AP↗
arXiv2008

Sobolev spaces on Lie manifolds and regularity for polyhedral domains

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for Lie submanifolds allows us also to extend to regular open subsets of Lie manifolds the classical results on traces of functions in suitable Sobolev spaces. Our main application is a regularity result on polyhedral domains $\PP \subset \RR^3$ using the weighted Sobolev spaces $\Kond{m}a(\PP)$. In particular, we show that there is no loss of $\Kond{m}a$--regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify $\Kond{m}a(\PP)$ with the Sobolev spaces on $\PP$ associated to the metric $r_{\PP}^{-2} g_E$, where $g_E$ is the Euclidean metric and $r_{\PP}(x)$ is a smoothing of the Euclidean distance from $x$ to the set of singular points of $\PP$. A suitable compactification of the interior of $\PP$ then becomes a regular open subset of a Lie manifold. We also obtain the well-posedness of a non-standard boundary value problem on a smooth, bounded domain with boundary $\maO \subset \RR^n$ using weighted Sobolev spaces, where the weight is the distance to the boundary.

math.AP↗
arXiv2008

Semilinear Schrödinger Flows on Hyperbolic Spaces: Scattering in H^1

We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schrödinger equations \begin{equation*} \begin{cases} &(i\partial_t+Δ_\g)u=u|u|^{2σ}; &u(0)=ϕ, \end{cases} \end{equation*} on the hyperbolic spaces $\H^d$, $d\geq 2$, for exponents $σ\in(0,2/(d-2))$. The main unexpected conclusion is scattering to linear solutions in the case of small exponents $σ$; for comparison, on Euclidean spaces scattering in $H^1$ is not known for any exponent $σ\in(1/d,2/d]$ and is known to fail for $σ\in(0,1/d]$. Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

math.AP↗