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Philippe Laurencot

Publications and source records attributed to Philippe Laurencot.

At least 19 recordsLinked to original sources
arXiv2023

Well-posedness to the discrete collision-induced breakage equation and various properties of solutions

A discrete version of the nonlinear collision-induced breakage equation is studied. Existence of solutions is investigated for a broad class of unbounded collision kernels and daughter distribution functions, the collision kernel $a_{i,j}$ satisfiying $a_{i,j} \leq A i j$ for some $A>0$. More precisely, it is proved that given suitable conditions, there exists at least one mass-conserving solution for all times. A result on the uniqueness of solutions is also demonstrated under reasonably general conditions. Furthermore, the propagation of moments, differentiability, and the continuous dependence of solutions are established, along with some invariance properties and the large-time behaviour of solutions.

math.CA↗
arXiv2017

A constrained model for MEMS with varying dielectric properties

A semilinear parabolic equation with constraint modeling the dynamics of a microelectromechanical system (MEMS) is studied. In contrast to the commonly used MEMS model, the well-known pull-in phenomenon occurring above a critical potential threshold is not accompanied by a breakdown of the model, but is recovered by the saturation of the constraint for pulled-in states. It is shown that a maximal stationary solution exists and that saturation only occurs for large potential values. In addition, the existence, uniqueness, and large time behavior of solutions to the evolution equation are studied.

math.AP↗
arXiv2017

Heterogeneous dielectric properties in MEMS models

An idealized electrostatically actuated microelectromechanical system (MEMS) involving an elastic plate with a heterogeneous dielectric material is considered. Starting from the electrostatic and mechanical energies, the governing evolution equations for the electrostatic potential and the plate deflection are derived from the corresponding energy balance. This leads to a free boundary transmission problem due to a jump of the dielectric permittivity across the interface separating elastic plate and free space. Reduced models retaining the influence of the heterogeneity of the elastic plate under suitable assumptions are obtained when either the elastic's plate thickness or the aspect ratio of the device vanishes.

math.AP↗
arXiv2017

Some singular equations modeling MEMS

In the past fifteen years mathematical models for microelectromechanical systems (MEMS) have been the subject of several studies, in particular due to the interesting qualitative properties they feature. Still most research is devoted to an illustrative, but simplified model which is deduced from a more complex model when the aspect ratio of the device vanishes, the so-called vanishing (or small) aspect ratio model. The analysis of the aforementioned complex model involving a moving boundary has started only recently and an outlook of the results obtained so far in this direction is provided in this survey.

math.AP↗
arXiv2015

On a three-dimensional free boundary problem modeling electrostatic mems

We consider the dynamics of an electrostatically actuated thin elastic plate being clamped at its boundary above a rigid plate. The model includes the harmonic electrostatic potential in the three-dimensional time-varying region between the plates along with a fourth-order semilinear parabolic equation for the elastic plate deflection which is coupled to the square of the gradient trace of the electrostatic potential on this plate. The strength of the coupling is tuned by a parameter $λ$ proportional to the square of the applied voltage. We prove that this free boundary problem is locally well-posed in time and that for small values of $λ$ solutions exist globally in time. We also derive the existence of a branch of asymptotically stable stationary solutions for small values of $λ$ and non-existence of stationary solutions for large values thereof, the latter being restricted to a disc-shaped plate.

math.AP↗
arXiv2014

Self-similarity in a thin film Muskat problem

The large time behavior of non-negative weak solutions to a thin film approximation of the two-phase Muskat problem is studied. A classification of self-similar solutions is first provided: there is always a unique even self-similar solution while a continuum of non-symmetric self-similar solutions exist for certain fluid configurations. Despite this non-uniqueness, convergence of all non-negative weak solutions towards a self-similar solution is proved.

math.AP↗
arXiv2014

A variational approach to a stationary free boundary problem modeling MEMS

A variational approach is employed to find stationary solutions to a free boundary problem modeling an idealized electrostatically actuated MEMS device made of an elastic plate coated with a thin dielectric film and suspended above a rigid ground plate. The model couples a non-local fourth-order equation for the elastic plate deflection to the harmonic electrostatic potential in the free domain between the elastic and the ground plate. The corresponding energy is non-coercive reflecting an inherent singularity related to a possible touchdown of the elastic plate. Stationary solutions are constructed using a constrained minimization problem. A by-product is the existence of at least two stationary solutions for some values of the applied voltage.

math.AP↗
arXiv2014

Large time behavior for the fast diffusion equation with critical absorption

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption $$ \partial_{t}u-Δu^m+u^q=0 \quad \quad \hbox{in} \ (0,\infty)\times\real^N\, $$ with $m_c:=(N-2)_{+}/N < m < 1$ and $q=m+2/N$. Given an initial condition $u_0$ decaying arbitrarily fast at infinity, we show that the asymptotic behavior of the corresponding solution $u$ is given by a Barenblatt profile with a logarithmic scaling, thereby extending a previous result requiring a specific algebraic lower bound on $u_0$. A by-product of our analysis is the derivation of sharp gradient estimates and a universal lower bound, which have their own interest and hold true for general exponents $q > 1$.

math.AP↗
arXiv2014

Absence of gelation and self-similar behavior for a coagulation-fragmentation equation

The dynamics of a coagulation-fragmentation equation with multiplicative coagulation kernel and critical singular fragmentation is studied. In contrast to the coagulation equation, it is proved that fragmentation prevents the occurrence of the gelation phenomenon and a mass-conserving solution is constructed. The large time behavior of this solution is shown to be described by a self-similar solution. In addition, the second moment is finite for positive times whatever its initial value. The proof relies on the Laplace transform which maps the original equation to a first-order nonlinear hyperbolic equation with a singular source term. A precise study of this equation is then performed with the method of characteristics.

math.AP↗
arXiv2014

The time singular limit for a fourth-order damped wave equation for MEMS

We consider a free boundary problem modeling electrostatic microelectromechanical systems. The model consists of a fourth-order damped wave equation for the elastic plate displacement which is coupled to an elliptic equation for the electrostatic potential. We first review some recent results on existence and non-existence of steady-states as well as on local and global well-posedness of the dynamical problem, the main focus being on the possible touchdown behavior of the elastic plate. We then investigate the behavior of the solutions in the time singular limit when the ratio between inertial and damping effects tends to zero.

math.AP↗
arXiv2014

Bounded traveling waves for a thin film with gravity and insoluble surfactant

Lubrication equations for a surfactant-driven flow of a thin layer of fluid are considered with a singular surfactant-dependent surface tension. It is shown that there exists a bounded traveling wave solution which connects a fully surfactant-coated film to an uncoated film. The regularity of the traveling wave depends on whether or not surface diffusion of the insoluble surfactant is included.

math.AP↗
arXiv2013

A free boundary problem modeling electrostatic MEMS: II. nonlinear bending effects

Well-posedness of a free boundary problem for electrostatic microelectromechanical systems (MEMS) is investigated when nonlinear bending effects are taken into account. The model describes the evolution of the deflection of an electrically conductive elastic membrane suspended above a fixed ground plate together with the electrostatic potential in the free domain between the membrane and the fixed ground plate. The electrostatic potential is harmonic in that domain and its values are held fixed along the membrane and the ground plate. The equation for the membrane deflection is a parabolic quasilinear fourth-order equation, which is coupled to the gradient trace of the electrostatic potential on the membrane.

math.AP↗
arXiv2013

A free boundary problem modeling electrostatic MEMS: I. Linear bending effects

The dynamical and stationary behaviors of a fourth-order evolution equation with clamped boundary conditions and a singular nonlocal reaction term, which is coupled to an elliptic free boundary problem on a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes two positive parameters $λ$ and $\varepsilon$ related to the applied voltage and the aspect ratio of the device, respectively. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant. Existence of a stable steady state is shown for sufficiently small $λ$. Non-existence of steady states is also established when $\varepsilon$ is small enough and $λ$ is large enough (depending on $\varepsilon$).

math.AP↗
arXiv2013

Finite time singularity in a free boundary problem modeling MEMS

The occurrence of a finite time singularity is shown for a free boundary problem modeling microelectromechanical systems (MEMS) when the applied voltage exceeds some value. The model involves a singular nonlocal reaction term and a nonlinear curvature term accounting for large deformations.

math.AP↗
arXiv2013

A fourth-order model for MEMS with clamped boundary conditions

The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes a positive voltage parameter $λ$. It is shown that there is a threshold value $λ_*>0$ of the voltage parameter such that no radially symmetric stationary solution exists for $λ>λ_*$, while at least two such solutions exist for $λ\in (0,λ_*)$. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when $λ>λ_*$.

math.AP↗
arXiv2013

Sign-Preserving Property for Some Fourth-Order Elliptic Operators in One Dimension and Radial Symmetry

For a class of one-dimensional linear elliptic fourth-order equations with homogeneous Dirichlet boundary conditions it is shown that a non-positive and non-vanishing right-hand side gives rise to a negative solution. A similar result is obtained for the same class of equations for radially symmetric solutions in a ball or in an annulus. Several applications are given, including applications to nonlinear equations and eigenvalue problems.

math.AP↗
arXiv2013

Dynamics of a free boundary problem with curvature modeling electrostatic MEMS

The dynamics of a free boundary problem for electrostatically actuated microelectromechanical systems (MEMS) is investigated. The model couples the electric potential to the deformation of the membrane, the deflection of the membrane being caused by application of a voltage difference across the device. More precisely, the electrostatic potential is a harmonic function in the angular domain that is partly bounded by the deformable membrane. The gradient trace of the electric potential on this free boundary part acts as a source term in the governing equation for the membrane deformation. The main feature of the model considered herein is that, unlike most previous research, the small deformation assumption is dropped, and curvature for the deformation of the membrane is taken into account what leads to a quasilinear parabolic equation. The free boundary problem is shown to be well-posed, locally in time for arbitrary voltage values and globally in time for small voltages values. Furthermore, existence of asymptotically stable steady-state configurations is proved in case of small voltage values as well as non-existence of steady-states if the applied voltage difference is large. Finally, convergence of solutions of the free boundary problem to the solutions of the well-established small aspect ratio model is shown.

math.AP↗