Searcharxiv⌕ Search

arXiv subjects

Taoufik Hmidi

Publications and source records attributed to Taoufik Hmidi.

54 records · Page 3Linked to original sources
arXiv2013

General position of a projection and its image under a free unitary Brownian motion

Given an orthogonal projection $P$ and a free unitary Brownian motion $Y = (Y_t)_{t \geq 0}$ in a $W^{\star}$-non commutative probability space such that $Y$ and $P$ are $\star$-free in Voiculescu's sense, the main result of this paper states that $P$ and $Y_tPY_t^{\star}$ are in general position at any time $t$. To this end, we study the dynamics of the unitary operator $SY_tSY_t^{\star}$ where $S = 2P-1$. More precisely, we derive a partial differential equation for the Herglotz transform of its spectral distribution, say $μ_t$. Then, we provide a flow on the interval $[-1,1]$ in such a way that the Herglotz transform of $μ_t$ composed with this flow is governed by both the Herglotz transforms of the initial ($t=0$) and the stationary ($t = \infty)$ distributions. This fact allows to compute the weight that $μ_t$ assigns to $z=1$ leading to the main result. As a by-product, the weight that the spectral distribution of the free Jacobi process assigns to $x=1$ follows after a normalization. In the last part of the paper, we use combinatorics of non crossing partitions in order to analyze the term corresponding to the exponential decay $e^{-nt}$ in the expansion of the $n$-th moment of $SY_tSY_t^{\star}$.

math.PR↗
arXiv2013

On the global well-posedness for Euler equations with unbounded vorticity

In this paper, we are interested in the global persistence regularity for the 2D incompressible Euler equations in some function spaces allowing unbounded vorticities. More precisely, we prove the global propagation of the vorticity in some weighted Morrey-Campanato spaces and in this framework the velocity field is not necessarily Lipschitz but belongs to the log-Lipschitz class $L^αL,$ for some $α\in (0,1).$

math.AP↗
arXiv2012

On the spectral distribution of the free Jacobi process

In this paper, we are interested in the free Jacobi process starting at the unit of the compressed probability space where it takes values and associated with the parameter values $λ=1, θ=1/2$. Firstly, we derive a time-dependent recurrence equation for the moments of the process (valid for any starting point and all parameter values). Secondly, we transform this equation to a nonlinear partial differential one for the moment generating function that we solve when $λ= 1, θ=1/2$. The obtained solution together with tricky computations lead to an explicit expression of the moments which shows that the free Jacobi process is distributed at any time $t$ as $(1/4)(2+Y_{2t}+Y_{2t}^{\star})$ where $Y$ is a free unitary Brownian motion. This expression is recovered relying on enumeration techniques after proving that if $a$ is a symmetric Bernoulli random variable which is free from $\{Y, Y^{\star}\}$, then the distributions of $Y_{2t}$ and that of $aY_taY_t^{\star}$ coincide. We close the exposition by investigating the spectral distribution associated with general sets of parameter values.

math.SP↗
arXiv2012

Incompressible limit for the 2D isentropic Euler system with critical initial data

We study in this paper the low Mach number limit for the 2d isentropic Euler system with ill-prepared initial data belonging to the critical Besov space $B_{2,1}^2$. By combining Strichartz estimates with the special structure of the vorticity we prove that the lifespan goes to infinity as the Mach number goes to zero. We also prove the strong convergence in the space of the initial data $B_{2,1}^2$ of the incompressible parts to the solution of the incompressible Euler system. There are at least two main difficulties: the first one concerns the Beale-Kato-Majda criterion which is not known to work for rough regularities. However, the second one is related to the critical aspect of the quantity $\Vert(\textnormal{div}\,v_{\varepsilon},\nabla c_{\varepsilon})\Vert_{L_{T}^{1}L^\infty}$ which has the scale of $B_{2,1}^2$ in the space variable.

math.AP↗
arXiv2011

Low Mach number limit for the isentropic Euler system with axisymmetric initial data

This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is $H^s$ {with $s>\frac52$.} Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the {lifespan $T_ε$} of the solutions is bounded below by $\log\log\log\frac1ε$, where $ε$ denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system, when the parameter $ε$ goes to zero. In the second part of the paper we address the same problem but for the Besov critical regularity $B_{2,1}^{\frac52}$. This case turns out to be more subtle at least due to two facts. The first one is related to the Beale-Kato-Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate $L^1_TL^\infty$ for the acoustic parts $(\nablaΔ^{-1}\diver\vepsilon,\cepsilon)$: it scales in the space variables like the space of the initial data.

math.AP↗
arXiv2011

Spectral distribution of the free unitary Brownian motion: another approach

We revisit the description provided by Ph. Biane of the spectral measure of the free unitary Brownian motion. We actually construct for any $t \in (0,4)$ a Jordan curve $γ_t$ around the origin, not intersecting the semi-axis $[1,\infty[$ and whose image under some meromorphic function $h_t$ lies in the circle. Our construction is naturally suggested by a residue-type integral representation of the moments and $h_t$ is up to a Möbius transformation the main ingredient used in the original proof. Once we did, the spectral measure is described as the push-forward of a complex measure under a local diffeomorphism yielding its absolute-continuity and its support. Our approach has the merit to be an easy yet technical exercise from real analysis.

math.OA↗
arXiv2009

Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data

In this paper we prove the global well-posedness for a three-dimensional Boussinesq system with axisymmetric initial data. This system couples the Navier-Stokes equation with a transport-diffusion equation governing the temperature. Our result holds uniformly with respect to the heat conductivity coefficient $κ\geq 0$ which may vanish.

math.AP↗
arXiv2009

On a maximum principle and its application to logarithmically critical Boussinesq system

In this paper we study a transport-diffusion model with some logarithmic dissipations. We look for two kinds of estimates. The first one is a maximum principle whose proof is based on Askey theorem concerning characteristic functions and some tools from the theory of $C_0$-semigroups. The second one is a smoothing effect based on some results from harmonic analysis and sub-Markovian operators. As an application we prove the global well-posedness for the two-dimensional Euler-Boussinesq system where the dissipation occurs only on the temperature equation and has the form $\frac{\DD}{\log^α(e^4+\DD)}$, with $α\in[0,\frac12]$. This result improves the critical dissipation $(α=0)$ needed for global well-posedness which was discussed in [15].

math.AP↗
arXiv2009

Time-dependent delta-interactions for 1D Schrödinger Hamiltonians

The non autonomous Cauchy problem for time dependent 1D point interactions is considered. The regularity assumptions for the coupling parameter are accurately analyzed and show that the general results for non autonomous linear evolution equations in Banach spaces are far from being optimal. In the mean time, this article shows an unexpected application of paraproduct techniques, initiated by J.M. Bony for nonlinear partial differential equations, to a classical linear problem.

math-ph↗
arXiv2008

On the global well-posedness for the axisymmetric Euler equations

This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in critical Besov spaces $B_{p,1}^{1+3/p}$. In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity.

math.AP↗