arXiv subjects

Nader Masmoudi

Publications and source records attributed to Nader Masmoudi.

At least 19 recordsLinked to original sources

arXiv2026

Long-time behaviour of two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane

In this paper, we study the long-time behavior of solutions to the two-dimensional Navier-Stokes equations in the presence of Couette flow on the half plane with Navier-slip boundary conditions. We prove that the total vorticity will approach \begin{align*} -1+\frac{M_2(\omega_{0})}{\nu^{3/2}(1+t)^{5/2}} \bar{\Omega}\left( \frac{x}{\sqrt{\nu(1+t)^3}}, \frac{y}{\sqrt{\nu(1+t)}} \right), \end{align*} where $-1$ is the vorticity of the Couette flow and $\bar{\Omega}$ is the kernel of a Fokker-Planck type operator $\mathcal{L}=\partial_Y^2+\frac32 X\partial_X+\frac12 Y\partial_Y+\frac52-Y\partial_X$. In the proof, we introduce a new idea of studying the spectrum of such type operators with boundary.

math.AP↗

arXiv2026

Gradient blowup of smooth vacuum solutions to 1D compressible Euler equations

We consider the isentropic compressible Euler equations in the half-line which govern the motion of gaseous fluids in contact with stationary vacuum boundary. We construct a large class of solutions that are initially smooth and square-integrable, and which, in finite time, transition to $C^{1-\mu}$ regularity for $\mu \in [1/2,1)$ near the boundary, leading to the gradient blowup at the boundary. It is based on stability analysis of self-similar waiting time solutions \cite{JLN2025} recently constructed by the authors.

math.AP↗

arXiv2025

Towards stable AI systems for Evaluating Arabic Pronunciations

Modern Arabic ASR systems such as wav2vec 2.0 excel at word- and sentence-level transcription, yet struggle to classify isolated letters. In this study, we show that this phoneme-level task, crucial for language learning, speech therapy, and phonetic research, is challenging because isolated letters lack co-articulatory cues, provide no lexical context, and last only a few hundred milliseconds. Recogniser systems must therefore rely solely on variable acoustic cues, a difficulty heightened by Arabic's emphatic (pharyngealized) consonants and other sounds with no close analogues in many languages. This study introduces a diverse, diacritised corpus of isolated Arabic letters and demonstrates that state-of-the-art wav2vec 2.0 models achieve only 35% accuracy on it. Training a lightweight neural network on wav2vec embeddings raises performance to 65%. However, adding a small amplitude perturbation (epsilon = 0.05) cuts accuracy to 32%. To restore robustness, we apply adversarial training, limiting the noisy-speech drop to 9% while preserving clean-speech accuracy. We detail the corpus, training pipeline, and evaluation protocol, and release, on demand, data and code for reproducibility. Finally, we outline future work extending these methods to word- and sentence-level frameworks, where precise letter pronunciation remains critical.

cs.CL↗

arXiv2025

Neural operators struggle to learn complex PDEs in pedestrian mobility: Hughes model case study

This paper investigates the limitations of neural operators in learning solutions for a Hughes model, a first-order hyperbolic conservation law system for crowd dynamics. The model couples a Fokker-Planck equation representing pedestrian density with a Hamilton-Jacobi-type (eikonal) equation. This Hughes model belongs to the class of nonlinear hyperbolic systems that often exhibit complex solution structures, including shocks and discontinuities. In this study, we assess the performance of three state-of-the-art neural operators (Fourier Neural Operator, Wavelet Neural Operator, and Multiwavelet Neural Operator) in various challenging scenarios. Specifically, we consider (1) discontinuous and Gaussian initial conditions and (2) diverse boundary conditions, while also examining the impact of different numerical schemes. Our results show that these neural operators perform well in easy scenarios with fewer discontinuities in the initial condition, yet they struggle in complex scenarios with multiple initial discontinuities and dynamic boundary conditions, even when trained specifically on such complex samples. The predicted solutions often appear smoother, resulting in a reduction in total variation and a loss of important physical features. This smoothing behavior is similar to issues discussed by Daganzo (1995), where models that introduce artificial diffusion were shown to miss essential features such as shock waves in hyperbolic systems. These results suggest that current neural operator architectures may introduce unintended regularization effects that limit their ability to capture transport dynamics governed by discontinuities. They also raise concerns about generalizing these methods to traffic applications where shock preservation is essential.

cs.LG↗

arXiv2025

Waiting Time Solutions in gas dynamics

In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent $\ga\in(1,3)$ in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least $C^1$ regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only H\"{o}lder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.

math.AP↗

arXiv2025

Blow-up of the one-dimensional wave equation with quadratic spatial derivative nonlinearity

We investigate the blow-up dynamics of smooth solutions to the one-dimensional wave equation with a quadratic spatial derivative nonlinearity, motivated by its applications in Effective Field Theory (EFT) in cosmology. Despite its relevance, explicit blow-up solutions for this equation have not been documented in the literature. In this work, we establish the non-existence of smooth, exact self-similar blow-up solutions and construct a five-parameter family of generalized self-similar solutions exhibiting logarithmic growth. Moreover, we prove the asymptotic stability of these blow-up solutions. Our proof tackles several significant challenges, including the non-self-adjoint nature of the linearized operator, the presence of unstable eigenvalues, and, most notably, the treatment of non-compact perturbations. By substantially advancing Donninger's spectral-theoretic framework, we develop a robust methodology that effectively handles non-compact perturbations. Key innovations include the incorporation of the Lorentz transformation in self-similar variables, an adaptation of the functional framework in [Merle-Raphael-Rodnianski-Szeftel, Invent.Math., 2022], and a novel resolvent estimate. This approach is general and robust, allowing for straightforward extensions to higher dimensions and applications to a wide range of nonlinear equations.

math.AP↗

arXiv2025

Incompressible and vanishing vertical viscosity limit for the compressible Navier-Stokes system with Dirichlet boundary conditions

In this paper, we show the incompressible and vanishing vertical viscosity limits for the strong solutions to the isentropic compressible Navier-Stokes system with anistropic dissipation, in a domain with Dirichlet boundary conditions in the general setting of ill-prepared initial data. We establish the uniform regularity estimates with respect to the Mach number $\epsilon$ and the vertical viscosity $\nu$ so that the solution exists on a uniform time interval $[0,T_0]$ independent of these parameters. The key steps toward this goal are the careful construction of the approximate solution in the presence of both fast oscillations and two kinds of boundary layers together with the stability analysis of the remainder. In the process, it is also shown that the solutions of the compressible systems converge to those of the incompressible system with only horizontal dissipation, after removing the fast waves whose horizontal derivative is bounded in $L_{T_0}^2L_x^2$ by $\min\{1, (\epsilon/\nu)^{\frac14}\}.$

math.AP↗

arXiv2024

Well-Posedness and Regularity of the Heat Equation with Robin Boundary Conditions in the Two-Dimensional Wedge

Well-posedness and higher regularity of the heat equation with Robin boundary conditions in an unbounded two-dimensional wedge is established in an $L^{2}$-setting of monomially weighted spaces. A mathematical framework is developed which allows to obtain arbitrarily high regularity without a smallness assumption on the opening angle of the wedge. The challenging aspect is that the resolvent problem exhibits two breakings of the scaling invariance, one in the equation and one in the boundary condition.

math.AP↗

arXiv2024

Singularity formed by the collision of two collapsing solitons in interaction for the 2D Keller-Segel system

It is well-known that the two-dimensional parabolic-elliptic Keller-Segel system admits finite-time blowup solutions, which is the case if the initial density has total mass greater than $8\pi$. Several constructive examples of such solutions have been given, where for all of them a perturbed stationary state undergoes scale instability and collapses at a point, resulting in an $8\pi$-mass concentration. It was conjectured that singular solutions concentrating more than one soliton simultaneously could exist. We construct rigorously such a new blowup mechanism, where two stationary states are simultaneously collapsing and colliding, resulting in a $16\pi$-mass concentration at a single blowup point, and with a new blowup rate which corresponds to the formal prediction by Seki, Sugiyama and Vel\'azquez. We develop, for the first time, a robust framework to rigorously construct blowup solutions that simultaneously involve the non-radial collision and concentration of several solitons, which we expect to have applications to other evolution problems.

math.AP↗

arXiv2024

Boundary driven instabilities of Couette flows

In this article, we prove that the threshold of instability of the classical Couette flow in $H^s$ for large $s$ is $\nu^{1/2}$. The instability is completely driven by the boundary. The dynamic of the flow creates a Prandtl type boundary layer of width $\nu^{1/2}$ which is itself linearly unstable. This leads to a secondary instability which in turn creates a sub-layer.

math.AP↗

arXiv2024

Well-posedness of the Stokes equations on a wedge with Navier-slip boundary conditions

We consider the incompressible and stationary Stokes equations on an infinite two-dimensional wedge with non-scaling invariant Navier-slip boundary conditions. We prove well-posedness and higher regularity of the Stokes problem in a certain class of weighted Sobolev spaces. The novelty of this work is the occurrence of two different scalings in the boundary condition, which is not treated so far for the Stokes system in unbounded wedge-type domains. These difficulties are overcome by first constructing a variational solution in a second order weighted Sobolev space and subsequently proving higher regularity up to the tip of the wedge by employing an iterative scheme. We believe that this method can be used for other problems with variational structure and multiple scales.

math.AP↗

arXiv2024

Symmetry Breaking in Chemical Systems: Engineering Complexity through Self-Organization and Marangoni Flows

Far from equilibrium, chemical and biological systems can form complex patterns and waves through reaction-diffusion coupling. Fluid motion often interferes with these self-organized concentration patterns. In this study, we investigate the influence of Marangoni-driven flows inside a thin layer of fluid ascending the outer surfaces of hydrophilic obstacles on the spatio-temporal dynamics of chemical waves in the modified Belousov-Zhabotinsky reaction. Our observations reveal that circular waves originate nearly simultaneously at the obstacles and propagate outward. In a covered setup, where evaporation is minimal, the wavefronts maintain their circular shape. However, in an uncovered setup with significant evaporation and resulting Marangoni flows, the interplay between surface tension-driven Marangoni flows and gravity destabilizes the wavefronts, creating distinctive flower-like patterns around the obstacles. Our analysis shows that here solutal Marangoni forces are more relevant than thermal ones. Our experiments further show that the number of petals formed increases linearly with the obstacle's diameter, though a minimum diameter is required for these instabilities to appear. These findings demonstrate the potential to 'engineer' specific wave patterns, offering a method to control and direct reaction dynamics. This capability is especially important for developing microfluidic devices requiring precise control over chemical wave propagation.

physics.flu-dyn↗

arXiv2024

Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation

We investigate the blow-up dynamics for the $L^2$ critical two-dimensional Zakharov-Kuznetsov equation \begin{equation*} \begin{cases} \partial_t u+\partial_{x_1} (\Delta u+u^3)=0, \mbox{ } x=(x_1,x_2)\in \mathbb{R}^2, \mbox{ } t \in \mathbb{R}\\ u(0,x_1,x_2)=u_0(x_1,x_2)\in H^1(\mathbb{R}^2), \end{cases} \end{equation*} with initial data $u_0$ slightly exceeding the mass of the ground state $Q$. Employing methodologies analogous to the Martel-Merle-Raphael blow-up theory for $L^2$ critical equations, more precisely for the critical NLS equation and the quintic generalized Korteweg-de Vries equation, we categorize the solution behaviors into three outcomes: asymptotic stability, finite-time blow-up, or divergence from the soliton's vicinity. The construction of the blow-up solution involves the bubbling of the solitary wave which ensures the universal behavior and stability of the blow-up.

math.AP↗

arXiv2024

Mack modes in supersonic boundary layer

Understanding the transition mechanism of boundary layer flows is of great significance in physics and engineering, especially due to the current development of supersonic and hypersonic aircraft. In this paper, we construct multiple unstable acoustic modes so-called Mack modes, which play a crucial role during the early stage of transition in the supersonic boundary layer. To this end, we develop an inner-outer gluing iteration to solve a hyperbolic-elliptic mixed type and singular system.

math.AP↗

arXiv2023

Rigorous derivation of the leapfrogging motion for planar Euler equations

The main goal of this paper is to explore the leapfrogging phenomenon in the inviscid planar flows. We show for 2d Euler equations that under suitable constraints, four concentrated vortex patches leapfrog for all time. When observed from a translating frame of reference, the evolution of these vortex patches can be described as a non-rigid time periodic motion. Our proof hinges upon two key components. First, we desingularize the symmetric four point vortex configuration, which leapfrogs in accordance with Love's result \cite{Love1893}, by concentrated vortex patches. Second, we borrow some tools from KAM theory to effectively tackle the small divisor problem and deal with the degeneracy in the time direction. Our approach is robust and flexible and solves a long-standing problem that has remained unresolved for many decades.

math.AP↗

arXiv2023

Stability Analysis of a Non-Separable Mean-Field Games for Pedestrian Flow in Large Corridors

We investigate the existence and stability of small perturbations of constant states of the generalized Hughes model for pedestrian flow in an infinitely large corridor. We show that constant flows are stable under a condition on the density. Our findings indicates that when the density is less than half of the maximum density $\rho_{m}/2$, which is the Lasry-Lions monotonicity condition, we can control the perturbation and prove positive stability results for the nonlinear Generalized Hughes model. However, due to wave propagation phenomena, we are unable to provide an answer for stability results when the density is higher. Our approach involves constructing an explicit solution for the linear problem in Fourier analysis and demonstrating, through a fixed-point argument, how to construct the solution for the full nonlinear mean-field games system.

math.AP↗

arXiv2023

Non-separable Mean Field Games for Pedestrian Flow: Generalized Hughes Model

In this paper, we present a new generalized Hughes model designed to intelligently depict pedestrian congestion dynamics, allowing pedestrian groups to either navigate through or circumvent high-density regions. First, we describe the microscopic settings of the model. The corresponding optimization problems are deterministic and can be formulated by a closed-loop model predictive control strategy. This microscopic setup leads in the mean-field limit to the Generalized Hughes model which is a class of non-separable mean field games system, i.e., Fokker-Planck equation and viscous Hamilton-Jacobi Bellman equation are coupled in a forward-backward structure. We give an overview on the mean field games in connection to our intelligent fluid model. Therefore, we show the existence of weak solutions to the Generalized Hughes model and analyze the vanishing viscosity limit of weak solutions. Finally, we illustrate the generalized Hughes model with various numerical experiments.

math.AP↗

arXiv2023

Diffusive limits of the steady state radiative heat transfer system: Curvature effects

This paper is devoted to the diffusive limit of the nonlinear radiative heat transfer system with curved boundary domain (\textit{two dimensional disk}). The solution constructed in \cite{ghattassi2022convergence} by the leading order interior solution and the boundary layer corrections fails here to approximate the solutions in $L^\infty$ sense for the diffusive limit. The present paper aims to construct a geometric correction to the boundary layer problem and obtain a valid approximate solution in $L^\infty$ sense. The main tools to overcome the convergence problem, are to use matched asymptotic expansion techniques, fixed-point theorems, linear and nonlinear stability analysis of the boundary layer problem. In particular, the spectral assumption on the leading order interior solution, which was proposed for the flat case in \cite{Bounadrylayer2019GHM2}, is shown to be still valid which guarantee the stability of the boundary layer expansion with geometric corrections. Moreover, the convergence result established in \cite[Lemma 10]{ghattassi2022convergence} remain applicable for the approximate solution with geometric corrections.

math.AP↗