arXiv subjects

Ayman Rimah Said

Publications and source records attributed to Ayman Rimah Said.

8 recordsLinked to original sources

arXiv2026

A Unified Approach to Mixing and Regularity for Passive Scalar Transport by Sobolev Vector Fields

We develop a new framework for quantitative estimates of passive scalar transport along Sobolev vector fields in $W^{1,p}$, when $p>1$. Our approach is based on Christ-Journé singular integral estimates. We show (i) a new stability estimate which quantifies the dependence of the solution on specific frequencies of the initial data; (ii) a new exponential mixing bound in the full DiPerna-Lions well-posedness class; (iii) propagation of logarithmic Fourier regularity of the solution; (iv) quantitative convergence rates for the vanishing diffusivity and mollification limits; and (v) a logarithmic decay rate for the standard DiPerna-Lions commutator.

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arXiv2026

Generic small-scale creation in the two-dimensional Euler equation

The Cauchy problem for the two-dimensional incompressible Euler equation is globally well-posed for smooth initial data. In this paper, we show that for a dense $G_δ$ set of initial data, the solutions lose regularity in infinite time, thereby confirming a long-standing conjecture of Yudovich in the smooth setting.

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arXiv2025

Mixing Estimates for Passive Scalar Transport by $BV$ Vector Fields

We prove a quantitative mixing estimate for the Cauchy problem for transport along divergence-free vector fields with bounded variation. By developing a framework that quantifies Ambrosio's regularisation scheme, we derive the first explicit bounds on the mixing rate for general $BV$ vector fields. Our analysis reveals that tetration (repeated exponentiation) emerges in the mixing rate from the local nature of Ambrosio's regularisation.

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arXiv2024

Small scale creation of the Lagrangian flow in 2d perfect fluids

In this paper we prove that for all solutions of the 2d Euler equations with initial vorticity with finite Sobolev smoothness then an initial data dependent norm of the associated Lagrangian flow blows up in infinite time at least like $t^{\frac{1}{3}}$. This initial data dependent norm quantifies the exact $L^2$ decay of the Fourier transform of the solution. This adapted norm turns out to be the exact quantity that controls a low to high frequency cascade which we then show to be the quantitative phenomenon behind the Lyapunov construction by Shnirelman.

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arXiv2023

On paracomposition and change of variables in Paradifferential operators

In this paper we revisit the hypothesis needed to define the "paracomposition" operator, an analogue to the classic pull-back operation in the low regularity setting, first introduced by S. Alinhac in [3]. More precisely we do so in two directions. First we drop the diffeomorphism hypothesis. Secondly we give estimates in global Sobolev and Zygmund spaces. Thus we fully generalize Bony's classic paralinearasition theorem giving sharp estimates for composition in Sobolev and Zygmund spaces. In order to prove that the new class of operations benefits of symbolic calculus properties when composed by a paradifferential operator, we discuss the pull-back of pseudodifferential and paradifferential operators which then become Fourier Integral Operators. In this discussion we show that those Fourier Integral Operators obtained by pull-back are pseudodifferential or paradifferential operators if and only if they are pulled-back by a diffeomorphism that is a change of variable. We give a proof of the change of variables in paradifferential operators. Finally we study the cutoff defining paradifferential operators and it's stability by successive composition. It is known that the cutoff becomes worse after each composition, we give a slightly refined version of the cutoffs proposed by Hörmander in [14] for which give an optimal estimate on the support of the cutoff after composition.

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arXiv2023

Regularity results on the flow maps of periodic dispersive Burgers type equations and the Gravity-Capillary equations

In the first part of this paper we prove that the flow associated to a dispersive Burgers equation with a non local term of the form $|D|^{α-1} \partial_x u$, $α\in [1,+\infty[$ is Lipschitz from bounded sets of $H^s_0(\mathbb{T};\mathbb{R})$ to $C^0([0,T],H^{s-(2-α)^+}_0(\mathbb{T};\mathbb{R}))$ for $T>0$ and $s>\lceil \fracα{α-1}\rceil-\frac{1}{2}$, where $H^s_0$ are the Sobolev spaces of functions with $0$ mean value, proving that the result obtained in [37] is optimal on the torus. The proof relies on a paradifferential generalization of a complex Cole-Hopf gauge transformation introduced by T.Tao in [43] for the Benjamin-Ono equation. For this we prove a generalization of the Baker-Campbell-Hausdorff formula for flows of hyperbolic paradifferential equations and prove the stability of the class of paradifferential operators modulo more regular remainders, under conjugation by such flows. For this we prove a new characterization of paradifferential operators in the spirit of Beals [9]. In the second part of this paper we use a paradifferential version of the previous method to prove that a re-normalization of the flow of the one dimensional periodic gravity capillary equation is Lipschitz from bounded sets of $H^s$ to $C^0([0,T],H^{s-\frac{1}{2}})$ for $T>0$ and $s>3+\frac{1}{2}$. This proves that the result obtained in [37] is optimal for the water waves system.

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arXiv2022

A geometric proof of the Quasi-linearity of the water-waves system

In the first part of this paper we prove that the flow associated to the Burgers equation with a non local term of the form $\partial_x |D|^{α-1} u$ fails to be uniformly continuous from bounded sets of $H^s({\mathbb D})$ to $C^0([0,T],H^s({\mathbb D}))$ for $T>0$, $s>\frac{1}{2}+2$, $0\leq α<2$, ${\mathbb D}={\mathbb R} \ \text{or} \ {\mathbb T} $. Furthermore we show that the flow cannot be $C^1$ from bounded sets of $H^s({\mathbb D})$ to $C^0([0,T],H^{s-1+(α-1)^+ +ε}({\mathbb D}))$ for $ε>0$. We generalize this result to a large class of nonlinear transport-dispersive equations in any dimension, that in particular contains the Whitham equation and the paralinearization of the water waves system with and without surface tension. The current result is optimal in the sense that for $α=2$ and ${\mathbb D}={\mathbb T}$ the flow associated to the Benjamin-Ono equation is Lipschitz on function with $0$ mean value $H^s_0$. In the second part of this paper we apply this method to deduce the quasi-linearity of the water waves system, which is the main result of this paper.

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arXiv2022

On the Cauchy problem of dispersive Burgers type equations

We study the paralinearised weakly dispersive Burgers type equation: $$\partial_t u+T_u \partial_xu+\partial_x |D|^{α-1}u=0,\ α\in ]1,2[,$$ which contains the main non linear "worst interaction" terms, that is low-high interaction terms, of the usual weakly dispersive Burgers type equation: \[ \partial_t u+u\partial_x u+\partial_x |D|^{α-1}u=0,\ α\in ]1,2[, \] with $u_0 \in H^s({\mathbb D})$, where ${\mathbb D}={\mathbb T} \text{ or } {\mathbb R}$. Through a paradifferential complex Cole-Hopf type gauge transform we introduced in [42], we prove a new a priori estimate in $H^s({\mathbb D})$ under the control of $\left\Vert D^{2-α}\left(u^2\right)\right\Vert_{L^1_tL^{\infty}_x}$, improving upon the usual hyperbolic control $\left\Vert \partial_x u\right\Vert_{L^1_tL^\infty_x}$. Thus we eliminate the "standard" wave breaking scenario in case of blow up as conjectured in [31]. For $α\in ]2,3[$ we show that we can completely conjugate the paralinearised dispersive Burgers equation to a semi-linear equation of the form: $$\partial_t \left[T_{e^{iT_{p(u)}}}u\right]+ \partial_x |D|^{α-1}\left[T_{e^{iT_{p(u)}}}u\right]=T_{R(u)}u,\ α\in ]2,3[,$$ where $T_{p(u)}$ and $T_{R(u)}$ are paradifferential operators of order $0$ defined for $u\in L^\infty_t C^{(2-α)^+}_*$.

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