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International audience; In this paper, we prove some new results on operational second order differential equations of elliptic type with general Robin boundary conditions in a non-commutative framework. The study is developed in Hölder spaces under some natural assumptions generalizing those in [4]. We give necessary and sufficient conditions on the data to obtain a unique strict solution satisfying the maximal regularity property, see Theorems 1 and 2. This work completes the one given in [4] and [12].

International audience; We study the asymptotic dynamics of the lake equations in the following two cases, an island shrinking to a point and an emerging island. For both cases, we derive an asymptotic lake-type equation. In the former case, the asymptotic dynamics includes an additional Dirac mass in the vorticity. The main mathematical difficulty is that the equations are singular when the water depth vanishes. We provide new uniform estimates in weighted spaces for the related stream functions which will imply the compactness result.

Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some specific interface conditions between three-dimensional and one-dimensional parts. In the case of the heat equation these conditions ensure the continuity of the solution and continuity in average of the normal flux. However for some computational reasons the strong continuity condition for the solution may be undesirable. It may be preferable to replace it by more flexible condition of continuity in average over the interface cross section. In the present paper we introduce and justify this alternative junction condition allowing such discontinuity of the solution of both stationary and non-stationary problems. The closeness of solutions of these hybrid dimension problems to the solution of the fully three-dimensional setting is proved. At the discrete level, finite volume schemes are considered, an error estimate is established. The new version of the MAPDD is compared to the classical one via numerical tests. It shows better stability of the new scheme.

This article is dedicated to the long time behavior of a finite volume approximation of general symmetrizable linear hyperbolic system on a bounded domain. In the continuous case this problem is very difficult, and the $\omega $–limit set (namely the set of all the possible long time limits) may be large and complicated to depict if no dissipation is introduced. In this article we prove that in general, with a stable finite volume scheme, the discrete solution converges to a steady state when the time goes to infinity. This property is a direct consequence of the numerical dissipation mechanisms used for stabilizing the discretization. We apply this result for determining the long time limit for several stabilizations of the wave system, and perform a formal link with the low Mach number problem of the nonlinear Euler system. Numerical experiments with the wave system are performed for confirming the theoretical results obtained.

International audience; In this article, we give an in-depth analysis of the problem of optimising the total population size for a standard logistic-diffusive model. This optimisation problem stems from the study of spatial ecology and amounts to the following question: assuming a species evolves in a domain, what is the best way to spread resources in order to ensure a maximal population size at equilibrium? {In recent years, many authors contributed to this topic.} We settle here the proof of two fundamental properties of optimisers: the bang-bang one which had so far only been proved under several strong assumptions, and the other one is the fragmentation of maximisers. Here, we prove the bang-bang property in all generality using a new spectral method. The technique introduced to demonstrate the bang-bang character of optimizers can be adapted and generalized to many optimization problems with other classes of bilinear optimal control problems where the state equation is semilinear and elliptic. We comment on it in a conclusion section.Regarding the geometry of maximisers, we exhibit a blow-up rate for the $BV$-norm of maximisers as the diffusivity gets smaller: if $\Omega$ is an orthotope and if $m_\mu$ is an optimal control, then $\Vert m_\mu\Vert_{BV}\gtrsim \sqrt{\mu}$. The proof of this results relies on a very fine energy argument.

Many systems in life sciences have been modeled by reaction–diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism like impulsive reaction–diffusion equations is necessary to analyze them. While several works tackled the issue of traveling waves for monotone reaction–diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction–diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction–diffusion equations. Our first result deals with the existence of traveling waves for monotone systems of impulsive reaction–diffusion equations. Our second result tackles the computation of spreading speeds for monotone systems of impulsive reaction–diffusion equations. We apply our methodology to a planar system of impulsive reaction–diffusion equations that models tree–grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.

In this article, we propose in-depth analysis and characterisation of the optimisers of the following optimisation problem: how to choose the initial condition u 0 in order to maximise the spatial integral at a given time of the solution of the semilinear equation u t −Δu = f(u), under L ∞ and L 1 constraints on u 0? Our contribution in the present paper is to give a characterisation of the behaviour of the optimiser u ¯ 0 when it does not saturate the L ∞ constraints, which is a key step in implementing efficient numerical algorithms. We give such a characterisation under mild regularity assumptions by proving that in that case u ¯ 0 can only take values in the ‘zone of concavity’ of f. This is done using two-scale asymptotic expansions. We then show how well-known isoperimetric inequalities yield a full characterisation of maximisers when f is convex. Finally, we provide several numerical simulations in one and two dimensions that illustrate and exemplify the fact that such characterisations significantly improve the computational time. All our theoretical results are in the one-dimensional case and we offer several comments about possible generalisations to other contexts, or obstructions that may prohibit doing so.

We consider the inverse problem of determining an electromagnetic potential appearing in an infinite cylindrical domain from boundary measurements. More precisely, we prove the stable recovery of some general class of magnetic field and electric potential from boundary measurements. Assuming some knowledge of the unknown coefficients close to the boundary, we obtain also some results of stable recovery with measurements restricted to some portion of the boundary. Our approach combines construction of complex geometric optics solutions and Carleman estimates suitably designed for our stability results stated in an unbounded domain.

In this work we consider the Landau-de Gennes model for liquid crystals with an external electromagnetic field to model the occurrence of the saturn ring effect under the assumption of rotational equivariance. After a rescaling of the energy, a variational limit is derived. Our analysis relies on precise estimates around the singularities and the study of a radial auxiliary problem in regions, where a continuous director field exists. Studying the limit problem, we explain the transition between the dipole and saturn ring configuration and the occurrence of a hysteresis phenomenon, giving a rigorous explanation of what was conjectured previously by [H. Stark, Eur. Phys. J. B 10, 311-321 (1999)]., Comment: 42 pages, 6 figures

We construct modal outgoing Green's kernels for the simplified Galbrun's equation under spherical symmetry, in the context of helioseismology. The coefficients of the equation are C 2 functions representing the solar interior model S , complemented with an isothermal atmospheric model. We solve the equation in vectorial spherical harmonics basis to obtain modal equations for the different components of the unknown wave motions. These equations are then decoupled and written in Schrodinger form, whose coefficients are shown to be C 2 apart from at most two regular singular points, and to decay like a Coulomb potential at infinity. These properties allow us to construct an outgoing Green's kernel for each spherical mode. We also compute asymptotic expansions of coefficients up to order r − 3 as r tends to infinity, and show numerically that their accuracy is improved by including the contribution from the gravity although this term is of order r − 3 .

International audience; This paper aims at reviewing and analysing the method of reflections. The latter is an iterative procedure designed to linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary, this method is particularly well-suited to numerical solvers relying on integral representation formulas. For the parallel and sequential forms of the method appearing in the literature, we propose a general abstract formulation in a given Hilbert setting and interpret the procedure in terms of subspace corrections. We then prove the unconditional convergence of the sequential form and propose a modification of the parallel one that makes it unconditionally converging. An alternative proof of convergence is provided in a case which does not fit into the previous framework. We finally present some numerical tests.

International audience; Problems with sign-changing coefficients occur, for instance, in the study of transmission problems with metamaterials. In this work, we present and analyze a generalized finite element method in the spirit of the Localized Orthogonal Decomposition, that is especially efficient when the negative and positive materials exhibit multiscale features. We derive optimal linear convergence in the energy norm independently of the potentially low regularity of the exact solution. Numerical experiments illustrate the theoretical convergence rates and show the applicability of the method for a large class of sign-changing diffusion problems.

We consider the drift-diffusion equation $$\begin{aligned} u_t-\varepsilon \varDelta u+\nabla \cdot (u\ \nabla K*u)=0 \end{aligned}$$ in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselves to the model case $$K(x)=-|x|$$ . We quantify the mass concentration phenomenon, a genuinely nonlinear effect, for radially symmetric solutions of this equation for small diffusivity $$\varepsilon $$ studied in our previous paper (Biler et al. in J Differ Equ 271:1092–1108, 2021), obtaining sharp upper and lower bounds for Sobolev norms.

We investigate the traveling wave solutions of a three-species system involving a single predator and a pair of strong-weak competing preys. Our results show how the predation may affect this dynamics. More precisely, we describe several situations where the environment is initially inhabited by the predator and by either one of the two preys. When the weak competing prey is an aboriginal species, we show that there exist traveling waves where the strong prey invades the environment and either replaces its weak counterpart, or more surprisingly the three species eventually co-exist. Furthermore, depending on the parameters, we can also construct traveling waves where the weaker prey actually invades the environment initially inhabited by its strong competitor and the predator. Finally, our results on the existence of traveling waves are sharp, in the sense that we find the minimal wave speed in all those situations., Comment: Journal of Differential Equations, Elsevier, 2021

International audience; In this work we determine the second-order coefficient in a parabolic equation from the knowledge of a single final data. Under assumptions on the concentration of eigenvalues of the associated elliptic operator, and the initial state, we show the uniqueness of solution, and we derive a Lipschitz stability estimate for the inversion when the final time is large enough. The Lipschitz stability constant grows exponentially with respect to the final time, which makes the inversion ill-posed. The proof of the stability estimate is based on a spectral decomposition of the solution to the parabolic equation in terms of the eigenfunctions of the associated elliptic operator, and an ad hoc method to solve a nonlinear stationary transport equation that is itself of interest.

International audience; We prove the well-posedness of a dynamical perfect plasticity model under general assumptions on the stress constraint set and on the reference configuration. The problem is studied by combining both calculus of variations and hyperbolic methods. The hyperbolic point of view enables one to derive a class of dissipative boundary conditions, somehow intermediate between homogeneous Dirichlet and Neumann ones. By using variational methods, we show the existence and uniqueness of solutions. Then we establish the equivalence between the original variational solutions and generalized entropic-dissipative ones, derived from a weak hyperbolic formulation for initial-boundary value Friedrichs' systems with convex constraints.

International audience; We study the semilinear elliptic equation −∆u + g(u)σ = µ with Dirichlet boundary conditions in a smooth bounded domain where σ is a nonnegative Radon measure, µ a Radon measure and g is an absorbing nonlinearity. We show that the problem is well posed if we assume that σ belongs to some Morrey class. Under this condition we give a general existence result for any bounded measure provided g satisfies a subcritical integral assumption. We study also the supercritical case when g(r) = |r| q−1 r, with q > 1 and µ satisfies an absolute continuity condition expressed in terms of some capacities involving σ. 2010 Mathematics Subject Classification. 35 J 61; 31 B 15; 28 C 05 .

International audience; The source terms of the Baer-Nunziato model involve highly non-linear return to equilibrium terms. In order to perform numerical simulations of realistic situations, accounting for this relaxation effects is mandatory. Unfortunately, with the classical forms retained for these source termsin the literature, building efficient, robust and accurate numerical schemes is a tricky task. In this paper, we propose different non-classical forms for these source terms. As for the classical ones, they all agree with the second law of thermodynamics and they are thus associated with a growth of an entropy. The great advantage of some of these new forms of source terms is that they are more linear with respect to the conservative variables. Consequently, this allows to propose more robust, efficient and accurate numerical schemes, in particular when considering fractional step approaches for which source terms and convection terms are solved separately.

Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are ``stable'' solutions and tractable by variational methods and maximum principles, while hyperbolic solutions are ``unstable'' solutions of which the qualitative behavior is less known. We introduce a new implicit function theorem based on the parabolic maximal regularity, which is applicable to problems with loss of derivatives. Clarifying the spectral structure of the corresponding linearized operator by harmonic analysis, we prove the existence of foliated hyperbolic solutions as well as elliptic solutions in the same regularity class., Comment: 23 pages

International audience; Abstract We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous source terms lying in some negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.

International audience; This paper investigates the large time behaviour of a three species reaction-diffusion system, modelling the spatial invasion of two predators feeding on a single prey species. In addition to the competition for food, the two predators exhibit competitive interactions and, under some parameter condition, they can also be considered as two mutants. When mutations occur in the predator populations, the spatial spread of invasion takes place at a definite speed, identical for both mutants. When the two predators are not coupled through mutation, the spreading behaviour exhibits a more complex propagating pattern, including multiple layers with different speeds. In addition, some parameter conditions reveal situations where a nonlocal pulling phenomenon occurs and in particular where the spreading speed is not linearly determined.

This paper is concerned with the spatio-temporal dynamics of nonnegative bounded entire solutions of some reaction–diffusion equations in $$\mathbb {R}^N$$ in any space dimension N. The solutions are assumed to be localized in the past. Under certain conditions on the reaction term, the solutions are then proved to be time-independent or heteroclinic connections between different steady states. Furthermore, either they are localized uniformly in time, or they converge to a constant steady state and spread at large time. This result is then applied to some specific bistable-type reactions.

We study a variational Ginzburg-Landau type model depending on a small parameter $\varepsilon>0$ for (tangent) vector fields on a $2$-dimensional Riemannian manifold $S$. As $\varepsilon\to 0$, these vector fields tend to have unit length so they generate singular points, called vortices, of a (non-zero) index if the genus $\mathfrak{g}$ of $S$ is different than $1$. Our first main result concerns the characterization of canonical harmonic unit vector fields with prescribed singular points and indices. The novelty of this classification involves flux integrals constrained to a particular vorticity-dependent lattice in the $2\mathfrak{g}$-dimensional space of harmonic $1$-forms on $S$ if $\mathfrak{g}\geq 1$. Our second main result determines the interaction energy (called renormalized energy) between vortex points as a $\Gamma$-limit (at the second order) as $\varepsilon\to 0$. The renormalized energy governing the optimal location of vortices depends on the Gauss curvature of $S$ as well as on the quantized flux. The coupling between flux quantization constraints and vorticity, and its impact on the renormalized energy, are new phenomena in the theory of Ginzburg-Landau type models. We also extend this study to two other (extrinsic) models for embedded hypersurfaces $S\subset \mathbb{R}^3$, in particular, to a physical model for non-tangent maps to $S$ coming from micromagnetics., Comment: 71 pages, 1 figure

International audience; We study the chemotaxis model ∂ t u = div(∇u − u∇w) + θv − u in (0, ∞) × Ω, ∂ t v = u − θv in (0, ∞) × Ω, ∂ t w = D∆w − αw + v in (0, ∞) × Ω, with no-flux boundary conditions in a bounded and smooth domain Ω ⊂ R 2 , where u and v represent the densities of subpopulations of moving and static individuals of some species, respectively, and w the concentration of a chemoattractant. We prove that, in an appropriate functional setting, all solutions exist globally in time. Moreover, we establish the existence of a critical mass M c > 0 of the whole population u + v such that, for M ∈ (0, M c), any solution is bounded, while, for almost all M > M c , there exist solutions blowing up in infinite time. The building block of the analysis is the construction of a Liapunov functional. As far as we know, this is the first result of this kind when the mass conservation includes the two subpopulations and not only the moving one.

We introduce a mean field game model for pedestrians moving in a given domain and choosing their trajectories so as to minimize a cost including a penalization on the difference between their own velocity and that of the other agents they meet. We prove existence of an equilibrium in a Lagrangian setting by using its variational structure, and then study its properties and regularity.

International audience; The existence of weak solutions to the obstacle problem for a nonlocal semilinear fourth-order parabolic equation is shown, using its underlying gradient flow structure. The model governs the dynamics of a microelectromechanical system with heterogeneous dielectric properties.

The weak localization or enhanced backscattering phenomenon has received a lot of attention in the literature. The enhanced backscattering cone refers to the situation that the wave backscattered by a random medium exhibits an enhanced intensity in a narrow cone around the incoming wave direction. This phenomenon can be analyzed by a formal path integral approach. Here a mathematical derivation of this result is given based on a system of equations that describes the second-order moments of the reflected wave. This system derives from a multiscale stochastic analysis of the wave field in the situation with high-frequency waves and propagation through a lossy medium with fine scale random microstructure. The theory identifies a duality relation between the spreading of the wave and the enhanced backscattering cone. It shows how the cone, its regularity and width relate to the statistical structure of the random medium. We discuss how this information in particular can be used to estimate the internal structure of the random medium based on observations of the reflected wave.

International audience; The main goal of this paper is to investigate the boundary controllability of some coupled parabolic systems in the cascade form in the case where the boundary conditions are of Robin type. In particular, we prove that the associated controls satisfy suitable uniform bounds with respect to the Robin parameters, that let us show that they converge towards a Dirichlet control when the Robin parameters go to infinity. This is a justification of the popular penalisation method for dealing with Dirichlet boundary data in the framework of the controllability of coupled parabolic systems.

The goal of this paper is to derive rigorously macroscopic traffic flow models from microscopic models. More precisely, for the microscopic models, we consider follow-the-leader type models with different types of drivers and vehicles which are distributed randomly on the road. After a rescaling, we show that the cumulative distribution function converge to the solution of a macroscopic model. We also make the link between this macroscopic model and the so-called LWR model.

International audience; In this work, we consider a mathematical model of syntrophic relationship between two microbial species of the anaerobic digestion process including mortality (or decay) terms. We focus on the acetogenesis and hydrogenotrophic methanogenesis phases. Our study gives a quite comprehensive analysis of a syntrophic model by analyzing the joined effects of syntrophy relationship, mortality, substrate inhibition and input concentrations that were neglected in previous studies. Using a general class of growth rates, the necessary and sufficient conditions of existence and local stability of all steady states of the four-dimensional system are determined according to the operating parameters. This general model exhibits a rich behavior with the coexistence of two microbial species, the bi-stability, the multiplicity of coexistence steady states, and the existence of two steady states of extinction of the first species. The operating diagram shows how the model behaves by varying the control parameters and illustrates the effect of the inhibition and the new input substrate concentration (hydrogen) on the reduction of the coexistence region and the emergence of a bi-stability region. Similarly to the classical chemostat model, including the substrate inhibition can destabilize a two-tiered microbial 'food chain' where the stability depends on the initial condition.

We derive high order homogenized models for the incompressible Stokes system in a cubic domain filled with periodic obstacles. These models have the potential to unify the three classical limit problems (namely the ``unchanged' Stokes system, the Brinkman model, and the Darcy's law) corresponding to various asymptotic regimes of the ratio $\eta\equiv a_{\epsilon}/\epsilon$ between the radius $a_{\epsilon}$ of the holes and the size $\epsilon$ of the periodic cell. What is more, a novel, rather surprising feature of our higher order effective equations is the occurrence of odd order differential operators when the obstacles are not symmetric. Our derivation relies on the method of two-scale power series expansions and on the existence of a ``criminal' ansatz, which allows to reconstruct the oscillating velocity and pressure $(\u_{\epsilon},p_{\epsilon})$ as a linear combination of the derivatives of their formal average $(\u_{\epsilon}^{*},p_{\epsilon}^{*})$ weighted by suitable corrector tensors. The formal average $(\u_\epsilon^{*},p_{\epsilon}^{*})$ is itself the solution to a formal, infinite order homogenized equation, whose truncation at any finite order is in general ill-posed. Inspired by the variational truncation method of \cite{smyshlyaev2000rigorous,cherednichenko2016full}, we derive, for any $K\in\N$, a well-posed model of order $2K+2$ which yields approximations of the original solutions with an error of order $O(\epsilon^{K+3})$ in the $L^{2}$ norm. Furthermore, the error improves up to the order $O(\epsilon^{2K+4})$ if a slight modification of this model remains well-posed. Finally, we find asymptotics of all homogenized tensors in the low volume fraction limit $\eta\rightarrow 0$ and in dimension $d\>3$. This allows us to obtain that our effective equations converge coefficient-wise to either of the Brinkman or Darcy regimes which arise when $\eta$ is respectively equivalent, or greater than the critical scaling $\eta_{\mathrm{crit}}\sim\epsilon^{2/(d-2)}$

We prove existence of \begin{document}$ L^1 $\end{document} -weak solutions to the reaction-diffusion system obtained as a stationary version of the system arising for the evolution of concentrations in a reversible chemical reaction, coupled with space diffusion. This extends a previous result by the same authors where restrictive assumptions on the number of chemical species are removed.

This paper is dedicated to the analysis of a mesoscopic model which describes sedimentation of inertialess suspensions in a viscous flow at mesoscopic scaling. The paper is divided into two parts, the first part concerns the analysis of the transport-Stokes model including a global existence and uniqueness result for $L^1\cap L^\infty$ initial densities with finite first moment. We investigate in particular the case where the initial condition is the characteristic function of the unit ball and show that we recover Hadamard-Rybczynski result, that is, the spherical shape of the droplet is preserved in time. In the second part of this paper, we derive a surface evolution model in the case where the initial shape of the droplet is axisymmetric. We obtain a 1D hyperbolic equation including non local operators that are linked to the convolution formula with respect to the singular Green function of the Stokes equation. We present a local existence and uniqueness result and show that we recover the Hadamard-Rybczynski result as long as the modelling is well defined and finish with numerical simulations in the spherical case.

International audience; We investigate the numerical discretization of a two-stream kinetic system with an internal state, such system has been introduced to model the motion of cells by chemotaxis. This internal state models the intracellular methylation level. It adds a variable in the mathematical model, which makes it more challenging to simulate numerically. Moreover, it has been shown that the macroscopic or mesoscopic quantities computed from this system converge to the Keller-Segel system at diffusive scaling or to the velocity-jump kinetic system for chemotaxis at hyperbolic scaling. Then we pay attention to propose numerical schemes uniformly accurate with respect to the scaling parameter. We show that these schemes converge to some limiting schemes which are consistent with the limiting macroscopic or kinetic system. This study is illustrated with some numerical simulations and comparisons with Monte Carlo simulations.

International audience; In this work, we study the exponential stability of the stationary distribution of a McKean-Vlasov equation, of nonlinear hyperbolic type which was recently derived in \cite{de_masi_hydrodynamic_2015,fournier_toy_2016}. We complement the convergence result proved in \cite{fournier_toy_2016} using tools from dynamical systems theory. Our proof relies on two principal arguments in addition to a Picard-like iteration method. First, the linearized semigroup is positive which allows to precisely pinpoint the spectrum of the infinitesimal generator. Second, we use a time rescaling argument to transform the original quasilinear equation into another one for which the nonlinear flow is differentiable. Interestingly, this convergence result can be interpreted as the existence of a locally exponentially attracting center manifold for a hyperbolic equation.

Well-posedness classes for degenerate elliptic problems in \begin{document}$ {\mathbb R}^N $\end{document} under the form \begin{document}$ u = \Delta {{\varphi}}(x,u)+f(x) $\end{document} , with locally (in \begin{document}$ u $\end{document} ) uniformly continuous nonlinearities, are explored. While we are particularly interested in the \begin{document}$ L^\infty $\end{document} setting, we also investigate about solutions in \begin{document}$ L^1_{loc} $\end{document} and in weighted \begin{document}$ L^1 $\end{document} spaces. We give some sufficient conditions in order that the uniqueness and comparison properties hold for the associated solutions; these conditions are expressed in terms of the moduli of continuity of \begin{document}$ u\mapsto {{\varphi}}(x,u) $\end{document} . Under additional restrictions on the dependency of \begin{document}$ {{\varphi}} $\end{document} on \begin{document}$ x $\end{document} , we deduce the existence results for the corresponding classes of solutions and data. Moreover, continuous dependence results follow readily from the existence claim and the comparison property. In particular, we show that for a general continuous non-decreasing nonlinearity \begin{document}$ {{\varphi}}: {\mathbb R}\mapsto {\mathbb R} $\end{document} , the space \begin{document}$ L^\infty $\end{document} (endowed with the \begin{document}$ L^1_{loc} $\end{document} topology) is a well-posedness class for the problem \begin{document}$ u = \Delta {{\varphi}}(u)+f(x) $\end{document} .

Moving bottlenecks in road traffic represent an interesting mathematical problem, which can be modeled via coupled PDE-ODE systems. We consider the case of a scalar conservation law modeling the evolution of vehicular traffic and an ODE with discontinuous right-hand side for the bottleneck. The bottleneck usually corresponds to a slow moving vehicle influencing the bulk traffic flow via a moving flux pointwise constraint. The definition of solutions requires a special entropy condition selecting nonclassical shocks and we prove existence of such solutions for initial data with bounded variation. Approximate solutions are constructed via the wave-front tracking method and their limit are solutions of the Cauchy problem PDE-ODE.

International audience; We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with Lipschitz continuous topography. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment ∆x of the L 2 norm of the gradient of approximate solutions. By Diperna's method we conclude the strong convergence towards bounded weak entropy solutions.

We focus on the spreading properties of solutions of monostable equations with non-linear diffusion. We consider both the porous medium diffusion and the fast diffusion regimes. Initial data may have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity may involve a weak Allee effect, which tends to slow down the process. We study the balance between these three effects (nonlin-ear diffusion, initial tail, KPP nonlinearity/Allee effect), revealing the separation between "no acceleration" and "acceleration". In most of the cases where acceleration occurs, we also give an accurate estimate of the position of the level sets., arXiv admin note: text overlap with arXiv:1505.04626

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[ \Delta^m u = \pm u^{\alpha} \quad \text{ in } \mathbb R^n \] with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants $n$, $m$ and $\alpha$. Remarkably, we are able to provide necessary and sufficient conditions on the exponent $\alpha$ to guarantee the existence of such solutions in $\mathbb R^n$. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive., Comment: 18 pages, 0 figure

In this paper we propose a notion of irreversibility for the evolution of cracks in the presence of cohesive forces, which allows for different responses in the loading and unloading processes, motivated by a variational approximation with damage models, and we investigate its applicability to the construction of a quasi-static evolution in a simple one-dimensional model. The cohesive fracture model arises naturally via $$\Gamma $$ -convergence from a phase-field model of the generalized Ambrosio-Tortorelli type, which may be used as regularization for numerical simulations.

Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result for when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations., SCOPUS: ar.j, info:eu-repo/semantics/published

We consider the elliptic quasilinear equation - Δ m ⁢ u = u p ⁢ | ∇ ⁡ u | q {-\Delta_{m}u=u^{p}\lvert\nabla u\rvert^{q}} in ℝ N {\mathbb{R}^{N}} , q ≥ m {q\geq m} and p > 0 {p>0} , 1 < m < N {1 . Our main result is a Liouville-type property, namely, all the positive C 1 {C^{1}} solutions in ℝ N {\mathbb{R}^{N}} are constant. We also give their asymptotic behaviour; all the solutions in an exterior domain ℝ N ∖ B r 0 {\mathbb{R}^{N}\setminus B_{r_{0}}} are bounded. The solutions in B r 0 ∖ { 0 } {B_{r_{0}}\setminus\{0\}} can be extended as continuous functions in B r 0 {B_{r_{0}}} . The solutions in ℝ N ∖ { 0 } {\mathbb{R}^{N}\setminus\{0\}} has a finite limit l ≥ 0 {l\geq 0} as | x | → ∞ {\lvert x\rvert\to\infty} . Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman-type estimate for the equation satisfied by the gradient.

We prove that the critical pulled front of Lotka-Volterra competition systems is nonlinearly asymptotically stable. More precisely, we show that perturbations of the critical front decay algebraically with rate t − 3 / 2 in a weighted L ∞ space. Our proof relies on pointwise semigroup methods and utilizes in a crucial way that the faster decay rate t − 3 / 2 is a consequence of the lack of an embedded zero of the Evans function at the origin for the linearized problem around the critical front.

We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space L 1 (R n), and more generally in L p (R n). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case n = 2. This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as t $\rightarrow$ +$\infty$. MSC2010: 35F55, 35L65. Notations. We denote $\times$ p the norm in Lebesgue L p (R n). The space of bounded measure over R m is M (R m) and its norm is denoted $\times$ M. The Dirac mass at X $\in$ R n is $\delta$ X or $\delta$ x=X. If $\nu$ $\in$ M (R m) and $\mu$ $\in$ M (R q), then $\nu$ $\otimes$ $\mu$ is the measure over R m+q uniquely defined by $\nu$ $\otimes$ $\mu$, $\psi$ = $\nu$, f $\mu$, g whenever $\psi$(x, y) $\not\equiv$ f (x)g(y). The closed halves of the real line are denoted R + and R --. * U.M.P.A., UMR CNRS-ENSL \# 5669. 46 all{\'e}e d'Italie

International audience; This work aims is to study a nonlinear second-order boundary value differential elliptic problem in one dimension where the nonlinearity concerns the solution and its first derivative. We assume that the source term can be non-smooth and the nonlinearity can grow faster than quadratic. First, we show the existence of a non-negative weak solution if we assume the existence of a super-solution. Second, we present a numerical algorithm to compute an approximation of the non-negative weak solution. The proposed algorithm is decomposed in two steps, the first one is devoted to computing a super-solution, and in the second one, the algorithm computes a sequence of solutions of an intermediate problem obtained by using the Yosida approximation of the nonlinearity. This sequence converges to the non-negative weak solution of the nonlinear equation. The numerical method is an application of the Newton method to the discretized version of the problem, but at each iteration, the resulting system can be indefinite. To overcome this difficulty, we introduce an adaptive non-overlapping domain decomposition method.

International audience; This paper is devoted to a generalisation of the quantum adiabatic theorem to a nonlinear setting. We consider a Hamiltonian operator which depends on the time variable and on a finite number of parameters and acts on a separable Hilbert space of which we select a fixed basis. We study an evolution equation in which this Hamiltonian acts on the unknown vector, while depending on coordinates of the unknown vector in the selected basis, thus making the equation nonlinear. We prove existence of solutions to this equation and consider their asymptotics in the adiabatic regime, i.e. when the Hamiltonian is slowly varying in time. Under natural spectral hypotheses, we prove the existence of normalised time dependent vectors depending on the Hamiltonian only, called instantaneous nonlinear eigenvectors, and show the existence of solutions which remain close to these vectors, up to a rapidly oscillating phase, in the adiabatic regime. We first investigate the case of bounded operators and then exhibit a set of spectral assumptions under which the result extends to unbounded Hamiltonians.

This paper deals with collisionless transport equationsin bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega $, orthogonallyinvariant velocity measure $\bm{m}(\d v)$ with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing andincoming fluxes. Under very general conditions, such equations are governedby stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $%L^{1}(\Omega \times V,\d x \otimes \bm{m}(\d v)).$ We give a general criterion of irreducibility of $%\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ and we show that, under very natural assumptions, if an invariant densityexists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ converges strongly (notsimply in Cesar\`o means) to its ergodic projection. We show also that if noinvariant density exists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ is\emph{sweeping} in the sense that, for any density $\varphi $, the total mass of $%U_{\mathsf{H}}(t)\varphi $ concentrates near suitable sets of zero measure as $%t\rightarrow +\infty .$ We show also a general weak compactness theoremwhich provides a basis for a general theory on existence of invariantdensities. This theorem is based on a series of results on smoothness andtransversality of the dynamical flow associated to $\left( U_{\mathsf{H}}(t)\right) _{t\geq0}.$

International audience; This works deals with sensitivity analysis for the Navier-Stokes equations. The aim is to provide an estimate of the variance of the velocity field when some of the parameters are uncertain and then to use the variance to compute confidence intervals for the output of the model. First, we introduce the physical model and analyse its stability. The sensitivity equations are derived, and their stability analysed as well. We propose a finite element-volume numerical scheme for the state and the sensitivity, which is integrated into the open-source industrial code TrioCFD. Finally, we present some numerical results: a steady and an unsteady test case for the channel flow problem are investigated. For the steady case, we compare the results to the Monte Carlo method and show how the sensitivity analysis technique succeeds in providing very accurate estimates of the variance. For the unsteady case, a new filtering procedure is proposed to deal with a sensitivity that grows in time. The filtered sensitivity is then used to compute the variance of the output and to provide confidence intervals.

We study the kernel function of the operator u $\rightarrow$ L $\mu$ u = --$\Delta$u + $\mu$ |x| 2 u in a bounded smooth domain $\Omega$ $\subset$ R N + such that 0 $\in$ $\partial$$\Omega$, where $\mu$ $\ge$ -- N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L $\mu$ u = 0 in $\Omega$ with boundary data $\nu$ + k$\delta$ 0 , where $\nu$ is a Radon measure on $\partial$$\Omega$ \ {0}, k $\in$ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L $\mu$ u = 0 in $\Omega$.