arXiv subjects

Mostafa Meliani

Publications and source records attributed to Mostafa Meliani.

12 recordsLinked to original sources

arXiv2026

The $hp$-FEM does not suffer from the pollution effect for piecewise-smooth Helmholtz problems with Gevrey regularity at boundaries

We consider the $hp$-FEM applied to the Helmholtz scattering problem with wavenumber $k$, truncated with a perfectly-matched layer. The scatterer consists of a combination of Dirichlet, Neumann, and penetrable obstacles together with variable coefficients. Provided that the Helmholtz solution operator is polynomially bounded in $k$, all coefficients are piecewise smooth, all boundary surfaces are Gevrey and all coefficients restricted to boundary surfaces are Gevrey together with all their normal derivatives, we show that the $hp$-FEM is quasioptimal when $p\geq 1+\varepsilon \log k$ and $hk/p$ is sufficiently small; i.e., the $hp$-FEM does not suffer from the pollution effect. This result generalises the analogous results in both [Bernkopf, Chaumont-Frelet, Melenk 2025] (proved for piecewise analytic coefficients and analytic boundaries) and [Galkowski, Lafontaine, Spence, Wunsch 2024] (proved for smooth coefficients that are analytic near analytic obstacles) to a much larger class of scatterers.

math.NA↗

arXiv2026

Well-posedness of a first-order formulation for fractionally damped nonlinear acoustics

In this work, we study the well-posedness of a quasilinear first-order-in-time system with memory arising in nonlinear acoustics. The model features a memory kernel describing the fractional damping and covers, as special cases, first-order formulations of Kuznetsov- and Westervelt-type equations. For completely monotone convolution kernels, we prove that the Westervelt-type system admits unique local-in-time solutions in bounded domains for space dimensions $d\leq3$, under homogeneous Dirichlet boundary conditions, suitable regularity, and smallness assumptions. The analysis is based on energy estimates exploiting novel nonlinear coercivity properties of the memory terms. Under additional assumptions on the resolvent of the kernel, we show that the smallness conditions can be significantly relaxed. For finite sums of exponential kernels our results generalize to the Kuznetsov-type system. Additionally, we show that the inviscid case (absence of memory kernel) in $\R^d$ can be treated by standard hyperbolic arguments in the case of the Kuznetsov-type system.

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arXiv2025

Blow-up criterion for the compressible Navier--Stokes system with inflow-outflow boundary conditions

We consider the compressible Navier-Stokes system in three dimensions with general inflow-outflow boundary conditions, meaning that we prescribe a boundary velocity which has non-zero normal component and accordingly the density is prescribed on the inflow part of the boundary. We establish a blow-up criterion in a class of strong solutions in the $L^p-L^q$ framework. In particular assuming the boundedness of the quantities $(\varrho^{-1}, \bu)$ and of a suitable norm of $\nabla_x \varrho$ the solution remains regular and the blow-up does not occur. We develop the condition on $\nabla_x \varrho$ because we need a new approach in order to accommodate the inhomogeneous boundary conditions, as the standard estimates on the material time derivative works when the normal component of the boundary velocity is zero.

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arXiv2025

Global existence for a fractionally damped nonlinear Jordan--Moore--Gibson--Thompson equation

In nonlinear acoustics, higher-order-in-time equations arise when taking into account a class of thermal relaxation laws in the modeling of sound wave propagation. In the literature, these families of equations came to be known as Jordan--Moore--Gibson--Thompson (JMGT) models. In this work, we show the global existence of solutions relying only on minimal assumptions on the nonlocal damping kernel. In particular, our result covers the until-now open question of global existence of solutions for the fractionally damped JMGT model with quadratic gradient nonlinearity. The factional damping setting forces us to work with non-integrable kernels, which require a tailored approach in the analysis to control. This approach relies on exploiting the specific nonlinearity structure combined with a weak damping provided by the nonlocality kernel.

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arXiv2025

$L^{p}-L^{q}$ existence for the open compressible MHD system

We study the local existence of solutions to the magnetohydrodynamics (MHD) system describing the motion of a compressible, viscous, electrically and heat conducting fluid in the $L^p-L^q$ class with inhomogeneous boundary conditions. The open system is allowed to receive incoming matter from the outside through (part of) the boundary which we refer to as an inflow boundary. This setup brings about a difficulty in estimating the regularity of the density $\varrho$ which we remedy by assuming appropriate hypotheses on the velocity field, domain boundary and on the boundary and initial data of $\varrho$. The main result ensures the local well-posedness of the full MHD system which is shown through a linearization combined with a Banach fixed-point theorem.

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arXiv2024

Well-posedness and global extensibility criteria for time-fractionally damped Jordan--Moore--Gibson--Thompson equation

In this paper, we consider the Jordan--Moore--Gibson--Thompson with a time-fractional damping term of the type $\delta \textup{D}_t^{1-\alpha} \Delta \psit$ where we allow the challenging so-called critical case ($\delta=0$). This equation arises in the context of acoustic propagation through thermally relaxed media. We tackle the question of long-time existence of the solution. More precisely, the goal of the paper is twofold: First, we establish local well-posedness of the initial boundary value problem, where we also provide a lower bound on the final time of existence as a function of initial data. Second, we prove a regularity result which guarantees, under the hypothesis of boundedness of certain quantities, that the local solution can be extended to be global-in-time.

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arXiv2024

Asymptotic-preserving hybridizable discontinuous Galerkin method for the Westervelt quasilinear wave equation

We discuss the asymptotic-preserving properties of a hybridizable discontinuous Galerkin method for the Westervelt model of ultrasound waves. More precisely, we show that the proposed method is robust with respect to small values of the sound diffusivity damping parameter $\delta$ by deriving low- and high-order energy stability estimates, and \emph{a priori} error bounds that are independent of $\delta$. Such bounds are then used to show that, when $\delta \rightarrow 0^+$, the method remains stable and the discrete acoustic velocity potential $\psi_h^{(\delta)}$ converges to $\psi_h^{(0)}$, where the latter is the singular vanishing dissipation limit. Moreover, we prove optimal convergence rates for the approximation of the acoustic particle velocity variable $\boldsymbol{v} = \nabla \psi$. The established theoretical results are illustrated with some numerical experiments.

math.NA↗

arXiv2023

Energy decay of some multi-term nonlocal-in-time Moore--Gibson--Thompson equations

This paper aims to explore the long-term behavior of some nonlocal high-order-in-time wave equations. These equations, which have come to be known as Moore--Gibson--Thompson equations, arise in the context of acoustic wave propagation when taking into account thermal relaxation mechanisms in complex media such as human tissue. While the long-term behavior of linear local-in-time acoustic equations is well understood, their nonlocal counterparts still retain many mysteries. We establish here a set of assumptions that ensures exponential decay of the energy of the system. These assumptions are then shown to be verified by a large class of rapidly decaying memory kernels. Under weaker assumptions on the kernel we show that one may still obtain that the energy vanishes but without a rate of convergence. Furthermore, we refine previous results on the local well-posedness of the studied equation and establish a necessary initial-data compatibility condition for the solvability of the problem.

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arXiv2023

A unified analysis framework for generalized fractional Moore--Gibson--Thompson equations: Well-posedness and singular limits

In acoustics, higher-order-in-time equations arise when taking into account a class of thermal relaxation laws in the modeling of sound wave propagation. In this work, we analyze initial boundary value problems for a family of such equations and determine the behavior of solutions as the relaxation time vanishes. In particular, we allow the leading term to be of fractional type. The studied model can be viewed as a generalization of the well-established (fractional) Moore--Gibson--Thompson equation with three, in general nonlocal, convolution terms involving two different kernels. The interplay of these convolutions will influence the uniform analysis and the limiting procedure. To unify the theoretical treatment of this class of local and nonlocal higher-order equations, we relax the classical assumption on the leading-term kernel and consider it to be a Radon measure. After establishing uniform well-posedness with respect to the relaxation time of the considered general model, we connect it, through a delicate singular limit procedure, to fractional second-order models of linear acoustics.

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arXiv2022

Mixed approximation of nonlinear acoustic equations: Well-posedness and a priori error analysis

Accurate simulation of nonlinear acoustic waves is essential for the continued development of a wide range of (high-intensity) focused ultrasound applications. This article explores mixed finite element formulations of classical strongly damped quasilinear models of ultrasonic wave propagation; the Kuznetsov and Westervelt equations. Such formulations allow simultaneous retrieval of the acoustic particle velocity and either the pressure or acoustic velocity potential, thus characterizing the entire ultrasonic field at once. Using non-standard energy analysis and a fixed-point technique, we establish sufficient conditions for the well-posedness, stability, and optimal a priori errors in the energy norm for the semi-discrete equations. For the Westervelt equation, we also determine the conditions under which the error bounds can be made uniform with respect to the involved strong dissipation parameter. A byproduct of this analysis is the convergence rate for the inviscid (undamped) Westervelt equation in mixed form. Additionally, we discuss convergence in the $L^q(\Omega)$ norm for the involved scalar quantities, where $q$ depends on the spatial dimension. Finally, computer experiments for the Raviart--Thomas (RT) and Brezzi--Douglas--Marini (BDM) elements are performed to confirm the theoretical findings.

math.NA↗

arXiv2022

Limiting behavior of quasilinear wave equations with fractional-type dissipation

In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin--Pipkin type. Aiming at minimal assumptions on the involved memory kernels -- which we allow to be weakly singular -- we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.

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arXiv2021

Analysis of general shape optimization problems in nonlinear acoustics

In various biomedical applications, precise focusing of nonlinear ultrasonic waves is crucial for efficiency and safety of the involved procedures. This work analyzes a class of shape optimization problems constrained by general quasi-linear acoustic wave equations that arise in high-intensity focused ultrasound (HIFU) applications. Within our theoretical framework, the Westervelt and Kuznetsov equations of nonlinear acoustics are obtained as particular cases. The quadratic gradient nonlinearity, specific to the Kuznetsov equation, requires special attention throughout. To prove the existence of the Eulerian shape derivative, we successively study the local well-posedness and regularity of the forward problem, uniformly with respect to shape variations, and prove that it does not degenerate under the hypothesis of small initial and boundary data. Additionally, we prove H\"older-continuity of the acoustic potential with respect to domain deformations. We then derive and analyze the corresponding adjoint problems for several different cost functionals of practical interest and conclude with the expressions of well-defined shape derivatives.

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