Makram Hamouda

Indiana University Bloomington | IUB · Department of Mathematics

In this article, we investigate the asymptotic behavior of the Mindlin–Timoshenko system under the influence of nonlinear dissipation affecting the rotation angle equations. Initially, we provide a concise review of the system’s solution existence. Subsequently, we demonstrate that the energy associated with the solution of the Mindlin–Timoshenko s...

We investigate the large time behavior of the solutions to the nonlinear focusing Schr\"odinger equation with a time-dependent damping in the energy sub-critical regime. Under non classical assumptions on the unsteady damping term, we prove some scattering results in the energy space.

The main focus of this paper is to analyze the behavior of a numerical solution of the Timoshenko system coupled with Thermoelasticity and incorporating second sound effects. In order to address this target, we employ the Physics-Informed Neural Networks (PINNs) framework to derive an approximate solution for the system. Our investigation delves in...

We investigate the Cauchy problem for the nonlinear Schr\"odinger equation with a time-dependent linear damping term. Under non standard assumptions on the loss dissipation, we prove the blow-up in the inter-critical regime, and the global existence in the energy subcritical case. Our results generalize and improve the ones in [9, 11, 21].

The article is devoted to investigating the initial boundary value problem for the damped wave equation in the scale-invariant case with time-dependent speed of propagation on the exterior domain. By presenting suitable multipliers and applying the test-function technique, we study the blow-up and the lifespan of the solutions to the problem with d...

We study in this article the blow-up of solutions to a coupled semilinear wave equations which are characterized by linear damping terms in the \textit{scale-invariant regime}, time-derivative nonlinearities, mass terms and Tricomi terms. The latter are specifically of great interest from both physical and mathematical points of view since they all...

We investigate the Cauchy problem for the nonlinear Schr ̈odingerequation with a time-dependent linear damping term. Under non standard assumptions on the loss dissipation, we prove the blow-up in the inter-critical regime, and the global existence in the energy subcritical case. Our results generalize and improve the ones in [9, 11, 21].

This paper is devoted to the study of boundary layers for a system of equations inherited from a threshold model between the compressible and incompressible Navier-Stokes equations. Indeed, when the viscosity is small enough, which is for example the case for water or air flows, and the domain is bounded, we observe large variations of the solution...

We consider in this article the weakly coupled system of wave equations in the scale-invariant case and with time-derivative nonlinearities. Under the assumption of small initial data, we obtain a better characterization of the delimitation of the blow-up region by deriving a new candidate for the critical curve. More precisely, we enhance the resu...

An improvement of [18] on the blow-up region and the lifespan estimate of a weakly coupled system of wave equations with damping and mass in the scale-invariant case and with time-derivative nonlinearity is obtained in this article. Indeed, thanks to a better understanding of the dynamics of the solutions, we give here a better characterization of...

In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein–de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the li...

In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integr...

In this article, we consider the damped wave equation in the scale-invariant case with time-dependent speed of propagation, mass term and time derivative nonlinearity. More precisely, we study the blow-up of the solutions to the following equation: ( E ) u t t − t 2 m Δ u + μ t u t + ν 2 t 2 u = | u t | p , in R N × [ 1 , ∞ ) , that we associate wi...

We consider in this article the damped wave equation in the scale-invariant case with combined two nonlinearities as source term, namely |ut|p+|u|q, and with small initial data. Owing to a better understanding of the influence of the damping term (μ1+tut) in the global dynamics of the solution, we obtain a new interval for μ that we conjecture to b...

We study in this article the asymptotic behavior of the Mindlin-Timoshenko system subject to a nonlinear dissipation acting only on the equations of the rotation angles. First, we briefly recall the existence of the solution of this system. Then, we prove that the energy associated with the Mindlin-Timoshenko system fulfills a dissipation relations...

In this article, we consider the damped wave equation in the \textit{scale-invariant case} with time-dependent speed of propagation, mass term and time derivative nonlinearity. More precisely, we study the blow-up of the solutions to the following equation: $$ (E) \quad u_{tt}-t^{2m}\Delta u+\frac{\mu}{t}u_t+\frac{\nu^2}{t^2}u=|u_t|^p, \quad \mbox{...

In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the...

In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integr...

We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider \begin{displaymath} \d (Tr) \hspace{1cm} u_{tt}-t^{2m}\Delta u=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), \end{displaymath} with small initial data, where $m\ge0$.\\ For the problem $(Tr...

We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider $$ (Tr) \hspace{1cm} u_{tt}-t^{2m}\Delta u=|u_t|^p+|u|^q, \quad \mbox{in}\ \mathbb{R}^N\times[0,\infty),$$ with small initial data, where $m\ge0$.\\ For the problem $(Tr)$ with $m=0$, which corresp...

We are interested in this article in studying the damped wave equation with localized initial data, in the \textit{scale-invariant case} with mass term and two combined nonlinearities. More precisely, we consider the following equation: $$ (E) {1cm} u_{tt}-\Delta u+\frac{\mu}{1+t}u_t+\frac{\nu^2}{(1+t)^2}u=|u_t|^p+|u|^q, \quad \mbox{in}\ \mathbb{R}...

In this article, we study the blow‐up of the damped wave equation in the scale‐invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: u t t − Δ u + μ 1 + t u t = | u t | p + | u | q , in ℝ N × [ 0 , ∞ ) , with small initial data. For μ < N ( q − 1 ) 2 and μ ∈ (0, μ∗), where μ∗ > 0 is depending...

We consider in this article the weakly coupled system of wave equations in the \textit{scale-invariant case} and with time-derivative nonlinearities. Under the usual assumption of small initial data, we obtain an improvement of the delimitation of the blow-up region by obtaining a new candidate for the critical curve. More precisely, we enhance the...

We consider in this article the damped wave equation, in the \textit{scale-invariant case} with combined two nonlinearities, which reads as follows: \begin{displaymath} \d (E) \hspace{1cm} u_{tt}-\Delta u+\frac{\mu}{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), \end{displaymath} with small initial data.\\ Compared to our previous wo...

In this article, we study the blow-up of the damped wave equation in the \textit{scale-invariant case} and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}-\Delta u+\frac{\mu}{1+t}u_t=|u_t|^p+|u|^q, \quad \mbox{in}\ \R^N\times[0,\infty), $$ with small initial data.\\ For $\mu < \frac{N(q-1)}{2}$ an...

In this article, we consider a one-dimensional Timoshenko system subject to different types of dissipation (linear and nonlinear damping). Based on a combination between the finite element and the finite difference methods, we design a discretization scheme for the different Timoshenko systems under consideration. We first come up with a numerical...

We are interested in this article in studying the damped wave equation in the scale-invariant case with mass term and two combined non-linearities. More precisely, we consider the following equation: (E) utt − ∆u +µ ν2 ut + 1 + t (1 + t)u=2|ut|p + |u|q, in RN × [0, ∞), with small initial data. Under some assumptions on the mass and damping coeffici...

In this article, we consider a one-dimensional Timoshenko system subject to different types of dissipation (linear and nonlinear dampings). Based on a combination between the finite element and the finite difference methods, we design a discretization scheme for the different Timoshenko systems under consideration. We first come up with a numerical...

In this paper, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. We first investigate the strong stability of this system, then we devote our efforts to obtain the strong lower energy estimates using Alabau--Boussouira's energy comparison principle introduced in \cite{2} (see also \cite{alabau}). One of th...

In this article we study the boundary layers for the subcritical modes of the viscous Linearized Primitive Equations (LPEs) in a cube at small viscosity. The boundary layers include the parabolic boundary layers, ordinary boundary layers, and their interaction-corner layers. The boundary layer correctors are determined by a phenomenological study r...

This book is a fairly unique resource regarding the rigorous mathematical treatment of boundary layer problems. The explicit methodology developed in this book extends in many different directions the concept of correctors initially introduced by J. L. Lions, and in particular the lower- and higher-order error estimates of asymptotic expansions are...

The Navier-Stokes equations appear as a singular perturbation of the Euler equations in which the small parameter ɛ is the viscosity or inverse of the Reynolds number. In many cases the convergence of the solutions of the Navier-Stokes equations to those of the Euler equations remains an outstanding open problem of mathematical physics. The result...

In this article we propose a new formulation of the equations of the humid atmosphere with a multi-phase saturation generalizing thus the model studied in Temam and Wu (2015 J. Funct. Anal. 269 2187–221) and Temam and Wang 2015 System Modeling and Optimization (New York: Springer). More precisely, we consider the more realistic situation where the...

We establish the vanishing viscosity limit of the zero-mode of the linearized Primitive Equations in a cube. Our method is based on the explicit construction and estimates of the boundary layers. This result, together with that in [12, 15], allows us to conclude the vanishing viscosity limit of the linearized Primitive Equations in a cube.

In fluid dynamics, we often study the flow of liquids and gases inside a region enclosed by a rigid boundary or around such a region. Some interesting applications in this field include analyzing, e.g., the motion of air around airplanes or automobiles to increase the efficiency of motion, the flow of atmosphere and oceans to predict the weather, a...

In this chapter, we will study the extension of the results on singular perturbations to higher dimensions. In dimension d ≥ 2, new problems arise related to the geometry of the domain, and in particular whether the domain is sufficiently regular or it has corners. Even if the domain is smooth, some boundary layers occur which are due to the curvat...

The study of Singular Perturbation Problems (SPP) in dimension one has a great importance since the boundary layer problems are generally one-dimensional problems in the direction normal to the boundary and, as we will see throughout the chapters of this book, many higher dimensional problems (in terms of singular perturbations) will be reduced to...

Following the approach introduced in [JT14a, JT11, JT12], we consider in this chapter the convection-diffusion equations in a circular domain where two characteristic points appear. The singular behaviors may occur at these points depending on the behavior of the given data, that is the domain (unit circle D), and f; see (5.1). As explained below,...

In this chapter and in Chapter 5, we investigate the boundary layers of convection-diffusion equations in space dimension one or two, and discuss additional issues to further develop the analysis performed in the previous Chapters 1 and 2

In this chapter, we present some recent progresses, which are based on [GJT16], about the boundary layer analysis in a domain enclosed by a curved boundary.

We investigate in this article the boundary layers appearing for a fluid under moderate rotation when the viscosity is small. The fluid is modeled by rotating type Stokes equations known also as the Barotropric mode equations in the primitive equations theory. First we derive the correctors that describe the sharp variations at large Reynolds numbe...

Motivated by the study of the corner singularities in the so-called cavity flow, we establish in the first part of this article, the existence and uniqueness of solutions in L²(Ω)² for the Stokes problem in a domain, when is a smooth domain or a convex polygon. This result is based on a new trace theorem and we show that the trace of u can be arbit...

The article is devoted to prove the existence and regularity of the solutions of the 3D inviscid Linearized Primitive Equations (LPEs) in a channel with lateral periodicity. This was assumed in a previous work (Hamouda et al. in Discret Contin Dyn Syst Ser S 6(2):401–422, 2013) which is concerned with the boundary layers generated by the correspond...

In this article we study the boundary layers for the viscous Linearized Primitive Equations (LPEs) when the viscosity is small. The LPEs are considered here in a cube. Besides the usual boundary layers that we analyze here too, corner layers due to the interaction between the different boundary layers are also studied.

In this work, we consider a nonlinear vibrating Timoshenko system with thermoelasticity with second sound. We recall first the results of well-posdness and regularity and the asymptotic behavior of the energy obtained in \cite{Ayadi}. Then, we use a fourth order finite difference scheme to compute the numerical solutions and thus we show the energy...

Motivated by the study of the corner singularities in the so-called cavity flow, we establish in this article, the existence and uniqueness of solutions in $L^2(\Omega)^2$ for the Stokes problem in a domain $\Omega,$ when $\Omega$ is a smooth domain or a convex polygon. We establish also a trace theorem and show that the trace of $u$ can be arbitra...

Our aim in this article is to study the boundary layers appearing at small viscosity for the Stokes solutions in a square (Formula presented.). By considering the Stokes problem in a square, we theoretically investigate the case where parabolic boundary layers are present. Using some divergence-free correctors, the asymptotic expansion of the visco...

Our aim in this article is to study the Linearized Navier-Stokes (LNS) problem including an interior singularity of the source function. At small viscosity, in addition to the classical boundary layers, interior layers are then developed inside the domain due to the discontinuities appearing in the limit inviscid solution. Using boundary layer func...

In this article, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. We discuss the well-posedness and the regularity of Timoshenko solution using the semi-group theory. Moreover, we etablish an explicit and general decay results for a wide class of relaxating functions which depend on a stability number $\m...

We give an asymptotic expansion, with respect to the viscosity, which is considered here to be small, of the solutions of the 3D linearized primitive equations (PEs) in a channel with lateral periodicity. A rigorous convergence result, in some physically relevant space, is proven. This allows, among other consequences, to confirm the natural choice...

We consider the Navier-Stokes equations of an incompressible fluid in a three dimensional curved domain with permeable walls in the limit of small viscosity. Using a curvilinear coordinate system, adapted to the boundary, we construct a cor-rector function at order ε j , j = 0, 1, where ε is the (small) viscosity parameter. This allows us to obtain...

The goal of this article is to study the asymptotic behaviour of the solutions of linearized Navier–Stokes equations (LNSE), when the viscosity is small, in a general (curved) bounded and smooth domain in 3 with a characteristic boundary. To handle the difficulties due to the curvature of the boundary, we first introduce a curvilinear coordinate sy...

The goal of this article is to study the boundary layer of the heat equation with thermal diffusivity in a general (curved), bounded and smooth domain in Rd, d ≥ 2, when the diffusivity parameter ε is small. Using a curvilinear coordinate system fitting the boundary, an asymptotic expansion, with respect to ε, of the heat solution is obtained at al...

In this presentation, we are interested in the behavior of the solution of the primitive equations (PEs) when the viscosity is very small. The PEs are the central equations for the large scale motion of the ocean and the atmosphere. Beside the general interest of determining the behavior at small (realistic) viscosity of the PEs, there is another m...

We establish the asymptotic behavior, when the viscosity goes to zero, of the solutions of the Linearized Primitive Equations (LPEs) in space dimension 2. More precisely, we prove that the LPEs solution behaves like the corresponding inviscid problem solution inside the domain plus an explicit corrector function in the neighborhood of some parts of...

We prove the existence of a strong corrector for the linearized in-compressible Navier–Stokes solution on a domain with characteristic bound-ary. This case is different from the noncharacteristic case considered in [7] and somehow physically more relevant. More precisely, we show that the linearized Navier–Stokes solutions behave like the Euler sol...

In this article, we numerically study the regularity loss of the solutions of non-parametric minimal surfaces with non-zero boundary conditions. Parts of the boundaries have non-positive mean curvature. As expected from theoretical results in such geometry, we find that the solutions may or may not satisfy the boundary conditions depending upon the...

In this article, we consider the asymptotic analysis of the solutions of the Navier-Stokes problem, when the viscosity goes to zero; we consider the flow in a channel of R 3 , in the non-characteristic boundary case. More precisely, a complete asymptotic expansion, at all orders, is given in the linear case. For the full nonlinear Navier-Stokes sol...

We study the global existence of solutions of a family of equations related to some physical phenomena in dimension 3, including the usual incompressible Navier–Stokes equations with an external force. In the latter case of blow-up at TmaxTmax an estimate of this time.

In this article, we derive the first term in the asymptotic expansion of the regularised minimal surface solution in the radially symmetric case when the domain is a pair of concentric circles in ℝ2. General domains and time-dependent problems will be considered elsewhere.

Our aim in this article is to study singularly perturbed problems which display boundary layers in the interior of the domain. These interior boundary layers which supplement the usual boundary layers at the boundary, are generated by discontinuities in the data. Second-order linear elliptic one-dimensional and multi-dimensional problems are consid...

In this article we study non classical singular perturbation problems involving boundary layers in the interior of the domain. As usual, these problems contain a small parameter which produces, when this parameter approaches zero, classical boundary layers located at the boundary. If we moreover consider a singular source function, we produce also...

Rapporteurs: Paolo Marcellini & L.A. Peletier Membres de Jury: Pierre Fabrie, Patrick Gerard, Danielle Hilhorst, Alain Lichnewsky, Jean Michel Rakotoson et Roger Temam.