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Belkacem Said-Houari

Publications and source records attributed to Belkacem Said-Houari.

At least 19 recordsLinked to original sources
arXiv2026

Global well-posedness and Decay estimates for solutions of the Navier--Stokes--Maxwell system

We investigate the global solvability and the large-time asymptotic behavior of solutions to the Navier--Stokes--Maxwell system in two and three space dimensions. The system describes the interaction between a viscous incompressible fluid and an electromagnetic field through the Lorenz force and Ohm's law. The system exhibits a parabolic--hyperbolic coupling in which the dissipative Navier--Stokes system interacts with the hyperbolic Maxwell system. The Maxwell system is damped through Ohm's law with the Maxwell correction. For sufficiently small initial data with suitable Sobolev regularity, we establish the global well-posedness of strong solutions and derive optimal decay rates for the solution and its higher-order spatial derivatives. In addition, we show that the electric field decays faster by the extra factor $(1+t)^{-1/2}$ compared to the velocity and magnetic components. One of the key ingredients of our analysis is a detailed study of the linearized system, which reveals refined decay properties of the electromagnetic components. These linear estimates play a crucial role in the analysis of the two-dimensional case, allowing us to overcome the logarithmic loss arising from the borderline decay estimates. For the nonlinear system, our approach is based on a time-weighted energy method, specifically designed to capture the distinct dissipative properties of the fluid and electromagnetic components. A suitable compensating functional is introduced to exploit the coupling between the electric and magnetic fields and thereby recover the missing dissipation of the magnetic field. Combined with careful analysis of the nonlinear terms, these estimates yield our desired result.

math.AP↗
arXiv2026

Optimal Time-Decay of global solutions to the Navier-Stokes-Maxwell system

We prove the existence and uniqueness of global-in-time solutions to the Navier--Stokes--Maxwell (NSM) system for small initial data in the critical Fujita--Kato space $\dot H^{\frac{d}{2}-1}(\mathbb R^d)$, in any dimension $d\geq 3$. Under the additional assumption that the initial data belong to the Besov space $\dot B^{-\frac{d}{2}}_{2,\infty}(\mathbb R^d)$, we also establish the optimal decay rate $t^{-\frac{s}{2}-\frac{d}{4}}$ at infinity of the solution in $\dot H^s(\mathbb R^d)$, for any $s\in\left(-\frac{d}{2},\frac{d}{2}-1\right]$. This is achieved by constructing a Lyapunov functional that is equivalent to the pointwise energy on the Fourier side, and by combining it with an adaptation of the Fourier splitting method to establish its optimal decay rate. All the analysis carried out here - from the global existence theory to the study of the large-time behavior - is performed within a framework that is uniform with respect to the speed of light $c\in(0,\infty)$. In particular, in the non-relativistic limit $c\to\infty$, this allows us to recover the same results for the corresponding limiting magnetohydrodynamic system.

math.AP↗
arXiv2026

On the decay estimates of a nonlocal convection-diffusion Hamer system

We consider the multi-dimensional Hamer model for radiating gases in its coupled hyperbolic--elliptic formulation. By means of energy estimates, we establish the global well-posedness for small initial data in hybrid Besov spaces with distinct regularity exponents at low and high frequencies. This framework enables us to relax the regularity assumptions required in \cite{Duan_Klem_Zhu_2010,Duan_Ruan_Zhu_2012}. In addition, we establish optimal time-decay estimates for solutions with initial data in the critical Besov space $\dot{B}_{2,\infty}^{-d/2}(\mathbb{R}^d)$, thus extending previous results obtained under the stronger assumption $L^1(\mathbb{R}^d)$. We discuss the optimality of these decay rates and derive improved decay rates under a zero-mass cancellation condition, corresponding to initial data in the larger negative Besov space $\dot B^{-d/2-1}_{2,\infty}(\mathbb R^d)$.

math.AP↗
arXiv2025

Optimal Neumann boundary and distributed control of the Westervelt equation with time-fractional attenuation

Optimal control of nonlinear acoustic waves is relevant in many medical ultrasound technologies, ranging from cancer therapy to targeted drug delivery, where it can help guide the precise deposition of acoustic energy. In this work, we study Neumann boundary and distributed control problems for tracking a prescribed pressure field governed by the Westervelt equation with time-fractional dissipation. This model captures nonlinear ultrasonic wave propagation in biological media and accounts for the experimentally observed power-law attenuation. We begin by extending the existing well-posedness theory for time-fractional equations to include inhomogeneous Neumann boundary data used as control inputs, which requires constructing an appropriate data extension and regularization. Using these analytical results for the forward problem, we prove the existence of globally optimal controls and analyze the stability of the optimization problem with respect to perturbations in the target pressure field and to vanishing regularization parameters. Finally, we investigate the associated adjoint equation, which has state-dependent coefficients, and use it to derive first-order necessary optimality conditions.

math.OC↗
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.

math.AP↗
arXiv2024

Westervelt-based modeling of ultrasound-enhanced drug delivery

We investigate a nonlinear multiphysics model motivated by ultrasound-enhanced drug delivery. The acoustic pressure field is modeled by Westervelt's quasilinear wave equation to adequately capture the nonlinear effects in ultrasound propagation. The nonlocal attenuation characteristic for soft biological media is modeled by acoustic damping of the time-fractional type. Additionally, acoustic medium parameters are allowed to depend on the temperature of the medium. The wave equation is coupled to the nonlinear Pennes heat equation with a pressure-dependent source to account for ultrasound waves heating up the tissue. Finally, the drug concentration is obtained as the solution to an advection-diffusion equation with a pressure-dependent velocity. Toward gaining a rigorous understanding of this system, we set up a fixed-point argument in the analysis combined with devising energy estimates that can accommodate the time-fractional damping. The energy arguments are, in part, carried out by employing time-weighted test functions to reduce the regularity assumptions on the initial temperature. The analysis reveals that different smoothness of the initial pressure, temperature, and concentration fields is needed as well as smallness of the pressure-temperature data in order to ensure non-degeneracy of the system and establish well-posedness. Our theoretical considerations are complemented by a numerical investigation of the system under more realistic boundary conditions. The numerical experiments, performed in different computational scenarios, underline the importance of considering nonlinear effects when modeling ultrasound-targeted drug delivery.

math.AP↗
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 $δ\textup{D}_t^{1-α} Δ\psit$ where we allow the challenging so-called critical case ($δ=0$). 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.

math.AP↗
arXiv2024

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.

math.AP↗
arXiv2024

The Westervelt--Pennes--Cattaneo model: local well-posedness and singular limit for vanishing relaxation time

In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on a coupled system of the Westervelt equation and the hyperbolic Pennes bioheat equation (Westervelt--Pennes--Cattaneo model). Using the energy method together with a fixed point argument, we prove that our model is locally well-posed and does not degenerate under a smallness assumption on the pressure data in the Westervelt equation. In addition, we perform a singular limit analysis and show that the Westervelt--Pennes--Fourier model can be seen as an approximation of the Westervelt--Pennes--Cattaneo model as the relaxation parameter tends to zero. This is done by deriving uniform bounds of the solution with respect to the relaxation parameter.

math.AP↗
arXiv2024

Global existence and asymptotic behavior of the Westervelt--hyperbolic Pennes system

In this work, we investigate the global existence and asymptotic behavior of a mathematical model of nonlinear ultrasonic heating based on a coupled system of the Westervelt equation and the hyperbolic Pennes bioheat equation (Westervelt--Pennes--Cattaneo model). First, we prove that the solution exists globally in time, provided that the lower-order Sobolev norms of the initial data are considered to be small, while the higher-order norms can be arbitrarily large. This is done using a continuity argument together with some interpolation inequalities. Second, we prove an exponential decay of the solution under the same smallness assumptions on the initial data.

math.AP↗
arXiv2024

Local well-posedness of a coupled Jordan-Moore-Gibson-Thompson-Pennes model of nonlinear ultrasonic heating

In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on the Jordan-Moore-Gibson-Thompson equation (JMGT) with temperature-dependent medium parameters coupled to the semilinear Pennes equation for the bioheat transfer. The equations are coupled via the temperature in the coefficients of the JMGT equation and via a nonlinear source term within the Pennes equation, which models the absorption of acoustic energy by the surrounding tissue. Using the energy method together with a fixed point argument, we prove that our model is locally well-posed, provided that the initial data are regular, small in a lower topology and the final time is short enough.

math.AP↗
arXiv2022

Time-weighted estimates for the Blackstock equation in nonlinear ultrasonics

High frequencies at which ultrasonic waves travel give rise to nonlinear phenomena. In thermoviscous fluids, these are captured by Blackstock's acoustic wave equation with strong damping. We revisit in this work its well-posedness analysis. By exploiting the parabolic-like character of this equation due to strong dissipation, we construct a time-weighted energy framework for investigating its local solvability. In this manner, we obtain the small-data well-posedness on bounded domains under less restrictive regularity assumptions on the initial conditions compared to the known results. Furthermore, we prove that such initial boundary-value problems for the Blackstock equation are globally solvable and that their solution decays exponentially fast to the steady state.

math.AP↗
arXiv2022

Global existence for the Jordan--Moore--Gibson--Thompson equation in Besov spaces

In this paper, we consider the Cauchy problem of a model in nonlinear acoustic, named the Jordan--Moore--Gibson--Thompson equation. This equation arises as an alternative model to the well-known Kuznetsov equation in acoustics. We prove global existence and optimal time decay of solutions in Besov spaces with a minimal regularity assumption on the initial data, lowering the regularity assumption required in \cite{Racke_Said_2019} for the proof of the global existence. Using a time-weighted energy method with the help of appropriate Lyapunov-type estimates, we also extend the decay rate in \cite{Racke_Said_2019} and show an optimal decay rate of the solution for initial data in the Besov space $\dot{B}_{2,\infty}^{-3/2}(\mathbb{R}^3)$, which is larger than the Lebesgue space $L^1(\R^3)$ due to the embedding $L^1(\mathbb{R}% ^3)\hookrightarrow \dot{B}_{2,\infty}^{-3/2}(\mathbb{R}^3)$. Hence we removed the $L^1$-assumption on the initial data required in \cite{Racke_Said_2019} in order to prove the decay estimates of the solution.

math.AP↗
arXiv2021

Local well-posedness of a coupled Westervelt-Pennes model of nonlinear ultrasonic heating

High-Intensity Focused Ultrasound (HIFU) waves are known to induce localized heat to a targeted area during medical treatments. In turn, the rise in temperature influences their speed of propagation. This coupling affects the position of the focal region as well as the achieved pressure and temperature values. In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on the Westervelt wave equation coupled to the Pennes bioheat equation that captures this so-called thermal lensing effect. We prove that this quasi-linear model is well-posed locally in time and does not degenerate under a smallness assumption on the pressure data.

math.AP↗
arXiv2021

Global well-posedness of the Cauchy problem for the Jordan--Moore--Gibson--Thompson equation with arbitrarily large higher-order Sobolev norms

In this paper, we consider the 3D Jordan--Moore--Gibson--Thompson equation arising in nonlinear acoustics. First, we prove that the solution exists globally in time provided that the lower order Sobolev norms of the initial data are considered to be small, while the higher-order norms can be arbitrarily large. This improves some available results in the literature. Second, we prove a new decay estimate for the linearized model and removing the $L^1$-assumption on the initial data. The proof of this decay estimate is based on the high-frequency and low-frequency decomposition of the solution together with an interpolation inequality related to Sobolev spaces with negative order.

math.AP↗
arXiv2020

Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation

Ultrasonic propagation through media with thermal and molecular relaxation can be modeled by third-order in time nonlinear wave-like equations with memory. This paper investigates the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan--Moore--Gibson--Thompson equation, in the so-called critical case, which corresponds to propagation in inviscid fluids. The memory has an exponentially fading character and type I, meaning that involves only the acoustic velocity potential. A major challenge in the global analysis is that the linearized equation's decay estimates are of regularity-loss type. As a result, the classical energy methods fail to work for the nonlinear problem. To overcome this difficulty, we construct appropriate time-weighted norms, where weights can have negative exponents. These problem-tailored norms create artificial damping terms that help control the nonlinearity and the loss of derivatives, and ultimately allow us to discover the model's asymptotic behavior.

math.AP↗
arXiv2020

Mathematical analysis of memory effects and thermal relaxation in nonlinear sound waves on unbounded domains

Motivated by the propagation of nonlinear sound waves through relaxing hereditary media, we study a nonlocal third-order Jordan-Moore-Gibson-Thompson acoustic wave equation. Under the assumption that the relaxation kernel decays exponentially, we prove local well-posedness in unbounded two- and three-dimensional domains. In addition, we show that the solution of the three-dimensional model exists globally in time for small and smooth data, while the energy of the system decays polynomially.

math.AP↗