Emmanuel Grenier

On the nonlinear instability of Euler and Prandtl equations

In this paper we give examples of nonlinearly unstable solutions of Euler equations in the whole space ℝ2, the half space ℝ × ℝ+, the periodic strip ℝ × , the strip ℝ × [−1,1], and the periodic torus 2, with an application to vortex sheets. Using the same methods, we prove an instability result for Prandtl-type boundary layers that appear in ℝ × ℝ+ and × ℝ+.

INCOMPRESSIBLE LIMIT FOR SOLUTIONS OF THE ISENTROPIC NAVIER-STOKES EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS

We study here the limit of global weak solutions of the compressible Navier-Stokes equations (in the isentropic regime) in a bounded domain, with Dirichlet boundary conditions on the velocity, as the Mach number goes to 0. We show that the velocity converges weakly in L 2 to a global weak solution of the incompressible Navier-Stokes equations. Moreover, the convergence in L 2 is strong under some geometrical assumption on.

Quasineutral limit of an euler-poisson system arising from plasma physics

In this paper, we study the quasineutral limit of an Euler-Poisson system arising from plasma physics i.e. the limit when the Debye length tends to of a non linear hyperbolic system coupled with a non linear elliptic equation.The proof uses pseudodiiferential energy estimates techniques, in order to justify classical limits in small time, for strong solutions.

Low Mach number limit of viscous compressible flows in the whole space

This paper is devoted to the low Mach number limit of weak solutions to the compressible Navier–Stokes equations for isentropic fluids in the whole space Rd (d = 2 or 3). This problem was investigated by P. L. Lions and N. Masmoudi. We present here a different approach based upon Strichartzs estimates for the linear wave equation in the inviscid case, which improves the convergence result and simplifies the proof. We prove that the velocity field is strongly compact and converges to a global weak solution of the incompressible Navier–Stokes equations.

Oscillations in quasineutral plasmas

The purpose of this article is to describe the limit, as the vacuum electric permittivity goes to zero, of a plasma physics system, deduced from the Vlasov-Poisson system for special initial data (distribution functions which are analytic in the space variable, with compact support in velocity), a limit also called {open_quotes}quasineutral regime{close_quotes} of the plasma, and the related oscillations of the electric field, with high frequency in time. 20 refs.

A Tumor Growth Inhibition Model For Low-Grade Glioma Treated With Chemotherapy or Radiotherapy

Purpose: To develop a tumor growth inhibition model for adult diffuse low-grade gliomas (LGG) able to describe tumor size evolution in patients treated with chemotherapy or radiotherapy. Experimental Design: Using longitudinal mean tumor diameter (MTD) data from 21 patients treated with first-line procarbazine, 1-(2-chloroethyl)-3-cyclohexyl-l-nitrosourea, and vincristine (PCV) chemotherapy, we formulated a model consisting of a system of differential equations, incorporating tumor-specific and treatment-related parameters that reflect the response of proliferative and quiescent tumor tissue to treatment. The model was then applied to the analysis of longitudinal tumor size data in 24 patients treated with first-line temozolomide (TMZ) chemotherapy and in 25 patients treated with first-line radiotherapy. Results: The model successfully described the MTD dynamics of LGG before, during, and after PCV chemotherapy. Using the same model structure, we were also able to successfully describe the MTD dynamics in LGG patients treated with TMZ chemotherapy or radiotherapy. Tumor-specific parameters were found to be consistent across the three treatment modalities. The model is robust to sensitivity analysis, and preliminary results suggest that it can predict treatment response on the basis of pretreatment tumor size data. Conclusions: Using MTD data, we propose a tumor growth inhibition model able to describe LGG tumor size evolution in patients treated with chemotherapy or radiotherapy. In the future, this model might be used to predict treatment efficacy in LGG patients and could constitute a rational tool to conceive more effective chemotherapy schedules. Clin Cancer Res; 18(18); 5071–80. ©2012 AACR.

Computational Modeling of Solid Tumor Growth: The Avascular Stage

In this paper, we present a mathematical model for avascular tumor growth and its numerical study in two and three dimensions. For this purpose, we use a multiscale model using PDEs to describe the evolution of the tumor cell densities. In our model, cell cycle regulation depends mainly on microenvironment. The cancer growth of volume induces cell motion and tumor expansion. According to biology, cells grow against a basal membrane which interacts mechanically with the tumor. We use a level set method to describe this membrane, and we compute its influence on cell movement, thanks to a Stokes equation. The evolution of oxygen, diffusing from blood vessels to cancer cells and used to estimate hypoxia, is given by a stationary diffusion equation solved with a penalty method. The model has been applied to investigate the therapeutic benefit of anti-invasive agents and constitutes now the basis of a numerical platform for tumor growth simulations.

Stabilizing effects of dispersion management

A cubic nonlinear Schrodinger equation (NLS) with periodically varying dispersion coefficient, as it arises in the context of fiber-optics communication, is considered. For sufficiently strong variation, corresponding to the so-called strong dispersion management regime, the equation possesses pulse-like solutions which evolve nearly periodically. This phenomenon is explained by constructing ground states for the averaged variational principle and justifying the averaging procedure. Furthermore, it is shown that in certain critical cases (e.g. quintic nonlinearity in one dimension and cubic nonlinearity in two dimensions) the dispersion management technique stabilizes the pulses which otherwise would be unstable. This observation seems to be new and is reminiscent of the well-known Kapitza’s effect of stabilizing the inverted pendulum by rapidly moving its pivot.

Defect measures of the vlasov-poisson system in the quasineutral regime

To study the quasi-neutral regime of a plasma, we introduce two defect measures with correctors in order to take the limit of the first two moments of the distribution function, in the time dependent case. Examples of non zero defect measures are also given.

Spectral stability of Prandtl boundary layers: An overview

Abstract In this paper we show how the stability of Prandtl boundary layers is linked to the stability of shear flows in the incompressible Navier–Stokes equations. We then recall classical physical instability results, and give a short educational presentation of the construction of unstable modes for Orr–Sommerfeld equations. We end the paper with a conjecture concerning the validity of Prandtl boundary layer asymptotic expansions.