Valérie Le Blanc
The aim of this paper is to study the $L^1$-stability of periodic stationary solutions of scalar convection-diffusion equations. We obtain dispersion in $L^2$ for all space dimensions using Kružkov type entropy. And when the space dimension is one, we estimate the number of sign changes of a solution to obtain $L^1$-stability.
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In this paper we establish the uniqueness of a solution to a stationary convection-diffusion equation in divergence form with an exponentially summable generalized divergence-free drift.
We study the existence and uniqueness of a solution to a linear stationary convection-diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a s…
A singularly perturbed system for doubly diffusive convection equations, called the artificial compressible system, is considered on a two-dimensional infinite layer for a parameters range where the Hopf bifurcation occurs in the corresponding incompressible system. The spectrum of the linearized operator in a time periodic function space is investigated in detail near the bifurcation point when the singular perturbation parameter is small. The results of this paper are the basis of the study of the nonlinear Hopf bifurcation problem and the singular limit of the time periodic bifurcating solu…
We study in this paper the periodic homogenization problem related to a strongly nonlinear reaction-diffusion equation. Owing to the large reaction term, the homogenized equation has a rather quite different form which puts together both the reaction and convection effects. We show in a special case that, the homogenized equation is exactly of a convection-diffusion type. The study relies on a suitable version of the well-known two-scale convergence method.
We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) process for a population of interacting particles crossing a domain with obstacle. Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective mo…
We prove existence results for a stationary Schr{ö}dinger equation with periodic magnetic potential satisfying a local integrability condition on the whole space using a critical value function.
This is a habilitation self-report describing results on forced periodic solutions of partial differential equations. Here averaging principle in the spirit of Bogoliubov and Mitropolsky for ODE's and Henry for parabolic equations are used together with topological methods such as fixed point index and topological degree.
First, the solution uniqueness and existence of a stationary anisotropic (linear) Stokes system with constant viscosity coefficients in a compressible framework on $n$-dimensional flat torus are analysed in a range of periodic Sobolev (Bessel-potential) spaces. By employing the Leray-Schauder fixed point theorem, the linear results are employed to show existence of solution to the stationary anisotropic (non-linear) Navier-Stokes incompressible system on torus in a periodic Sobolev space. Then the solution regularity results for stationary anisotropic Navier-Stokes system on torus are establis…
A singular perturbation problem from the artificial compressible system to the incompressible system is considered for a doubly diffusive convection when a Hopf bifurcation from the motionless state occurs in the incompressible system. It is proved that the Hopf bifurcation also occurs in the artificial compressible system for small singular perturbation parameter, called the artificial Mach number. The time periodic solution branch of the artificial compressible system is shown to converge to the corresponding bifurcating branch of the incompressible system in the singular limit of vanishing …
This article investigates the long-time behaviour of parabolic scalar conservation laws of the type $\partial_t u + \mathrm{div}_yA(y,u) - Δ_y u=0$, where $y\in\mathbb R^N$ and the flux $A$ is periodic in $y$. More specifically, we consider the case when the initial data is an $L^1$ disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between $u$ and the stationary solution vanishes for large times in $L^1$ norm. The proof uses a self-similar change of variables which is well-suited for the analysis of the long time behavi…
Main objective of this paper is to describe the dynamic transition of the incompressible MHD equations in a rectangular domain in $\mathbb{R}^{3}$. Our analysis shows that the system undergoes a first dynamic transition either to multiple equilibria or to periodic solutions. In the case of transition to multiple equilibria, the transition is a type-I (continuous) transition. In the case of transition to periodic solutions, the transition can be either type-I or type-II (jump).
We investigate the existence of ground states with fixed mass for the nonlinear Schrödinger equation with a pure power nonlinearity on periodic metric graphs. Within a variational framework, both the $L^2$-subcritical and critical regimes are studied. In the former case, we establish the existence of global minimizers of the NLS energy for every mass and every periodic graph. In the critical regime, a complete topological characterization is derived, providing conditions which allow or prevent ground states of a certain mass from existing. Besides, a rigorous notion of periodic graph is introd…
We study the homogenization of a steady diffusion equation in a highly heterogeneous medium made of two subregions separated by a periodic barrier through which the flow is proportional to the jump of the temperature by a layer conductance of the same order of magnitude of the materials in consideration. The macroscopic governing equations and the effective conductivity of the homogenized model are obtained by means of the two scale convergence technique. We show that under some hypothesis the homogenized systems contain convective terms of order one.
We consider the two-dimensional Euler equation with periodic boundary conditions. We construct time quasi-periodic solutions of this equation made of localized travelling profiles with compact support propagating over a stationary state depending on only one variable. The direction of propagation is orthogonal to this variable, and the support is concentrated on flat strips of the stationary state. The frequencies of the solution are given by the locally constant velocities associated with the stationary state.
Stationary Anisotropic Stokes, Oseen, and Navier-Stokes Systems: Periodic Solutions in $\R^n$
本文研究了可压缩框架下平稳各向异性 Stokes 和广义 Oseen 系统的周期解性质，并利用 Galerkin 算法将线性结果推广至不可压缩 Navier-Stokes 系统，证明了任意维度下解的存在性及特定维度下的唯一性与正则性。
We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport …
Stationary periodic solutions for nonlinear Dirac equations with non-coercive nonlinearity II: splitting
该研究针对具有Soler型非线性项的非线性狄拉克方程，通过分离圆因子降维，利用强制扰动与定量特征空间分离等方法，得到对应正谱阈值附近的非平凡周期解。
Modelling formation of stationary periodic patterns in growing population of motile bacteria
本文通过线性与非线性Fourier分析，研究趋化活性细菌增长种群中稳态周期图纹的形成条件及其振幅、波长等特征，并用数值模拟验证结果。
Stationary periodic solutions to Nonlinear Dirac equations with non-coercive potentials
本文研究了非线性狄拉克方程在非协化势下的稳态周期解问题，通过引入协化扰动和Sobolev范数估计，得到了非平凡解的存在性。
We discuss an approach to the computer assisted proof of the existence of branches of stationary and periodic solutions for dissipative PDEs, using the Brussellator system with diffusion and Dirichlet boundary conditions as an example, We also consider the case where the branch of periodic solutions emanates from a branch of stationary solutions through a Hopf bifurcation.
Hydrodynamic stability in the presence of a stochastic forcing:a case study in convection
本文研究瑞利-贝纳德对流中随机强迫对统计稳态导电态稳定性的影响，发现小强度强迫几乎无影响，强强迫具有去稳定效应，并提出结合严谨分析与计算的方法。
We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic-quintic complex Ginzburg-Landau equation about a periodically-stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg-Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically-stationary pulses in …
Stochastic-periodic Homogenization of Non-stationary Incompressible Navier-Stokes Type Equations
本文利用随机双尺度收敛方法，研究了各向异性非均匀介质上非平稳Navier-Stokes型方程变分公式的随机-周期均匀化，证明了极限方程与原变分形式一致。
Pattern formation in annular convection
本研究针对环形对流中的时空斑图形成，以大气对流和薄膜电对流为背景，利用O(2)-等变分岔理论预测稳态及时空斑图动力学。
Rayleigh-Bénard convection with stochastic forcing localised near the bottom
本文证明了在无限Prandtl数regime下，当随机扰动作用于底部薄层且noise的amplitude足够大时，三维Rayleigh-Bénard convection的随机流会稳定到唯一的stationary measure。
A unified spatiotemporal formulation with physics-preserving structure for time-dependent convection-diffusion problems
本文提出了一种统一的四维时空 formulation，用于时间依赖的对流-扩散问题，通过引入时间作为额外坐标并利用外微积分，实现了对物理结构的保持，并证明了其收敛性。
We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a finite-time blow-up. Finall…
In this technical report, we consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a f…
Large-time behavior of a family of finite volume schemes for boundary-driven convection-diffusion equations
本文研究了有限体积离散化对流-扩散方程及其系统在大时间行为下的特性，证明了多种线性和非线性模型中，有限体积方案保持离散解向离散稳态指数衰减的性质。
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