Alain MiranvilleUniversité de Poitiers | UP · Département de Mathématiques
Alain Miranville
Professor
Université de Poitiers (France), Henan Normal University and Xiamen University (P.R. China)
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277
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6,153
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Citations since 2017
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Publications (277)
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https://epubs.siam.org/doi/book/10.1137/1.9781611975925
Our aim in this paper is to prove the existence of global in time solutions for a model for the proliferative-to-invasive transition of hypoxic glioma cells. The equations consist of the coupling of a Cahn-Hilliard type equation for the tumor density and of a reaction-diffusion equation for the oxygen concentration. The main difficulty is to prove...
Our aim in this paper is to prove the existence of solutions to the Cahn–Hilliard equation with a general nonlinear source term. An essential difficulty is to obtain a global in time solution. Indeed, due to the presence of the source term, one cannot exclude the possibility of blow up in finite time when considering regular nonlinear terms and whe...
We consider a well-known model for the study of the evolution of an incompressible binary fluid flow in a two or three-dimensional bounded domain, through the fluid velocity u and an order parameter φ. This model consists of a system of two evolution equations, namely, the incompressible Navier-Stokes equations coupled with a convective Cahn-Hillia...
This paper is concerned with the determination and reduction for the 3D Brinkmann-Forchheimer equations, which gives the theory for numerical simulation and describes the complexity for fluid flow in medium porous. An improved Gronwall inequality is obtained for the proof of determining modes for 3D non-autonomous Brinkmann-Forchheimer equations su...
This paper is concerned with the Gromov-Hausdorff stability of global attractors for 3D Navier-Stokes equations with damping under variation of the domain. The Gromov-Hausdorff stability contains the Gromov-Hausdorff distances between two global attractors which may be in disjoint phase spaces, and the stability of semi-dynamical systems on global...
The numerical analysis of the coupled Cahn-Hilliard/Allen-Cahn system endowed with dynamic boundary conditions is studied in this article. We consider a semi-discretisation in space using a finite element method and we derive error estimates between the exact and the approximate solution. Then, using the backward Euler scheme for the time variable,...
Our aim in this paper is to study a mathematical model for high grade gliomas, taking into account lactates kinetics, as well as chemotherapy and antiangiogenic treatment. In particular, we prove the existence and uniqueness of biologically relevant solutions. We also perform numerical simulations based on different therapeutical situations that ca...
In this note we show that, under certain conditions on the coefficients, the solutions of the parabolic phase-lag heat conduction models are determined by an analytic semigroup for which the inner product determines the H2-norm of the temperature. As a consequence the H2-norm of the temperature decays in an exponential way.
A tight control of intracellular [Ca[Formula: see text]] is essential for the survival and normal function of cells. In this study we investigate key mechanistic steps by which calcium is regulated and calcium oscillations could occur using in silico modeling of membrane transporters. To do so we give a deterministic description of intracellular Ca...
Our aim in this paper is to study a coupled Cahn-Hilliard system for copolymer/homopolymer mixtures. We prove the existence, uniqueness and regularity of solutions. We then prove the existence of finite dimensional global attrac-tors.
This paper is concerned with the global solutions of the 3D compressible
micropolar fluid model in the domain to a subset of R3 bounded with
two coaxial cylinders that present the solid thermo-insulated walls, which is in
a thermodynamical sense perfect and polytropic. Compared with the classical
Navier-Stokes equations, the angular velocity w in t...
This paper is concerned with the tempered pullback dynamics of the two-dimensional Navier-Stokes equations with multi-delays defined in Lipschitz-like domain, subject to inhomogeneous Dirichlet boundary condition, which is a further research of Yang, Wang, Yan and Miranville (DCDS, 41(7), 2021, 3343-3366). In this presented work, we investigated th...
Our aim in this paper is to study a mathematical model for brain cancers with chemotherapy and antiangiogenic therapy effects. We prove the existence and uniqueness of biologically relevant (nonnegative) solutions. We then address the important question of optimal treatment. More precisely, we study the problem of finding the controls that provide...
Our aim in this paper is to study an optimal control problem for a tumor growth model. The state system couples an Allen-Cahn equation and a reaction diffusion equation that models the evolution of tumor in the presence of nutrient supply. Elimination of cancer cells via cytotoxic drug is considered and the concentration of the cytotoxic drug is re...
Altered metabolism, characterized by high concentration levels of lactate enzyme, contributes to tumor development, malignancy, and metastasis and introduces metabolic liabilities that can be employed in cancer treatment. Here, this paper aims to reach a desired lactate concentration, under the action of an optimal treatment dose, represented by a...
Our aim in this article is to study generalizations of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the associated system. In particular, we prove the existence of the g...
Our aim in this paper is to study a Cahn-Hilliard model with a symport term. This equation is proposed to model some energy mechanisms (e.g., lactate) in glial cells. The main difficulty is to prove the existence of a biologically relevant solution. This is achieved by considering a modified equation and taking a logarithmic nonlinear term. A secon...
The paper is concerned with a qualitative analysis for a nonlinear second-order parabolic problem, subject to non-homogeneous Cauchy–Stefan–Boltzmann boundary conditions, extending the types already studied. Under certain assumptions, we prove the existence, a priori estimates, regularity and uniqueness of a solution in the class Wp1,2(Q). Here we...
Our aim in this paper is to study an Allen-Cahn model based on a microforce balance and an unconstrained order parameter. We obtain the existence, uniqueness and regularity of solutions and prove that the solutions converge to those to the original Cahn-Hilliard equation on finite time intervals as a small parameter goes to zero.
This paper is concerned with the tempered pullback dynamics of the 2D Navier-Stokes equations with sublinear time delay operators subject to non-homogeneous boundary conditions in Lipschitz-like domains. By virtue of the estimates of background flow in Lipschitz-like domain and a new retarded Gronwall inequality, we establish the existence of pullb...
We consider a nonlinear delay evolution equation with multivalued perturbation on a noncompact interval. The nonlinearity, having convex and closed values, is upper hemicontinuous with respect to the solution variable. A basic question on whether there exists a solution set carrying $R_{\delta }$-structure remains unsolved when the operator familie...
Our aim in this paper is to study the existence and uniqueness of solutions to a Cahn-Hilliard type model proposed for image segmentation. We also prove the existence of unbounded (as time goes to infinity) solutions and give numerical simulations which illustrate our theoretical results.
In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen-Cahn/Cahn-Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.
This article is devoted to the analysis of the dynamics of a complex network of unstable reaction–diffusion systems. We demonstrate the existence of a non-empty parameter regime for which synchronization occurs in non-trivial attractors. We establish a lower bound of the dimension of the global attractor in an innovative manner, by proving a novel...
Interfaces play a key role on diseases development because they dictate the energy inflow of nutrients from the surrounding tissues. What is underestimated by existing mathematical models is the biological fact that cells are able to use different resources through nonlinear mechanisms. Among all nutrients, lactate appears to be a sensitive metabol...
Our aim in this paper is to prove the existence of solutions for a model for the proliferative-to-invasive transition of hypoxic glioma cells. The equations consist of the coupling of a reaction-diffusion equation for the tumor density and of a Cahn–Hilliard type equation for the oxygen concentration. The main difficulty is to prove the existence o...
In this note we study the problem proposed by the one-dimensional thermo-porous-elasticity of type II with quasi-static microvoids or, in mathematical terms, when the second time derivative of the volume fraction is so small that it can be negligible. It is known that the isothermal deformations decay in a slow way. Here we prove that the introduct...
Our aim in this article is to study the asymptotic behavior of a Cahn-Hilliard/ Allen-Cahn system coupled with a heat equation based on the type III heat conduction law with singular potentials. We also show further regularity results and we prove a strict separation property (from the pure states) in one space dimension.
In this paper, we first establish the existence of trajectory at-tractors for the 3D smectic-A liquid crystal flow system and 3D smectic-A liquid crystal flow-α model, and then prove that the latter trajectory attractor converges to the former one as the parameter α → 0 + .
Ebook available free of charge at
https://www.aimsciences.org/book/deds/volume/Volume%207
We study the long-time behavior, within the framework of infinite dimensional dynamical systems, of the Cahn–Hilliard equation endowed with a new class of dynamic boundary conditions. The system under investigation was recently derived by Liu–Wu (Arch Ration Mech Anal 233:167–247, 2019) via an energetic variational approach such that it naturally f...
We devise a first-order in time convex splitting scheme for a nonlocal Cahn-Hilliard-Oono
type equation with a transport term and subject to homogeneous Neumann boundary conditions. However, we prove the stability of our scheme when the time step is sufficiently small, according to the velocity field and the interaction kernel. Furthermore, we prov...
This paper is concerned with the large time behavior of uniform attractors for a 3D non-autonomous incompressible Navier-Stokes equations with nonlinearity in bounded domain which governing the motion of fluid flow. Based on the theory of strong and weak uniform attractors established by Chepyzhov and Vishik [11], Zelik [47], we present a new frame...
Our aim is to review the mathematical tools usefulness in MR data management for glioma diagnosis and treatment optimization. MRI does not give access to organs variations in hours or days. However a lot of multiparametric data are generated. Mathematics could help to override this paradox, the aim of this article is to show how. We first make a re...
This paper is concerned with the finite dimensional global attractor for 2D generalized quasi-geostrophic equations on unbounded domain in Lp(R2) with p > 2 . We first prove that the nonlocal operator generated by global solutions is sectorial in L2(R2). Then, we establish the existence of global attractor for the generalized 2D quasi-geostrophic e...
Our aim in this paper is to study ODEs models in view of applications to brain metabolites variations in the circadian rhythm. We address the well-posedness of the models, as well as the nonnegativity of the solutions. We then give numerical simulations which we compare with real medical data.
In this paper we consider the one-dimensional type II thermoviscoelastic theory with voids. We prove that generically we have exponential stability of the solutions. This is a striking fact if one compares it with the behavior in the case of the classical thermoviscoelastic theory based on the classical Fourier law for which the decay is genericall...
We consider the compressible Navier-Stokes-Cahn-Hilliard system describing the behavior of a binary mixture of compressible, viscous and macroscopically immiscible fluids. The equations are endowed with dynamic boundary conditions which allows taking into account the interaction between the fluid components and the rigid walls of the physical domai...
Our aim in this paper is to study a mathematical model for the proliferative-to-invasive transition of hypoxic glioma cells. We prove the existence and uniqueness of nonnegative solutions and then address the important question of whether the positive solutions undergo extinction or permanence. More precisely, we prove that this depends on the boun...
Our aim in this article is to study the well-posedness and properties of a system with delay which is related with brain glutamate and glutamine kinetics. In particular, we prove the existence and uniqueness of nonnegative solutions. We also give numerical simulations and compare their order of magnitude with experimental data.
The paper is concerned with a qualitative analysis for a nonlinear second-order parabolic problem, subject to non-homogeneous Cauchy–Neumann boundary conditions, extending the types already studied. Under some certain assumptions, we prove the existence, estimate, regularity and uniqueness of a classical solution. The considered nonlinear second-or...
We consider the problem of the long time dynamics for a diffuse interface model for tumor growth. The model describes the growth of a tumor surrounded by host tissues in the presence of a nutrient and consists in a Cahn-Hilliard-type equation for the tumor phase coupled with a reaction-diffusion equation for the nutrient concentration. We prove tha...
The aim of this article is to show how a tumor can modify energy substrates fluxes in the brain to support its own growth. To address this question we use a modeling approach to explain brain nutrient kinetics. In particular we set up a system of 17 equations for oxygen, lactate, glucose concentrations and cells number in the brain. We prove the ex...
Our aim in this article is to study generalizations of the noncon-served Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures for heat conduction and with logarithmic nonlinear terms. We obtain well-posedness results and study the asymptotic behavior of the system. In particular, we prove the existence of the global a...
In this paper we consider the one-dimensional type III thermoelastic theory with voids. We prove that generically we have exponential stability of the solutions. This is a striking fact if one compares it with the behavior in the case of the thermoelastic theory based on the classical Fourier law for which the decay is generically slower.
Our aim in this paper is to prove the existence of solutions to a Cahn-Hilliard type equation with a proliferation term and a logarithmic nonlinear term. Such an equation was proposed in view of biological applications. The main difficulty comes from the fact that we no longer have the conservation of the spatial average of the order parameter, con...
The aim of this paper is to study the time decay of the solutions for two models of the one-dimensional phase-lag thermoelasticity with two temperatures. The first one is obtained when the heat flux vector and the inductive temperature are approximated by a second-order and first-order Taylor polynomial, respectively. In this case, the solutions de...
We investigate the well-posedness and the stability of the solutions for several Taylor approximations of the phase-lag two-temperature equations. We give conditions on the parameters which guarantee the existence and uniqueness of solutions as well as the stability and the instability of the solutions for each approximation.
We consider a phase field model based on a generalization of the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Neu-mann boundary conditions. The originality here, compared with previous works, is that we obtain global in time and dissipative estimates, so that, in particular, we prove, in one and two space d...
We consider the problem of the long time dynamics for a diffuse interface model for tumor growth. The model describes the growth of a tumor surrounded by host tissues in the presence of a nutrient and consists in a Cahn-Hilliard-type equation for the tumor phase coupled with a reaction-diffusion equation for the nutrient concentration. We prove tha...
The motion of two contiguous incompressible and viscous fluids is described within the diffuse interface theory by the so-called Model H. The system consists of the Navier-Stokes equations, which are coupled with the Cahn-Hilliard equation associated to the Ginzburg-Landau free energy with physically relevant logarithmic potential. This model is st...
The aim of this article is to study the well-posedness and properties of a fast-slow system which is related with brain lactate kinetics. In particular, we prove the existence and uniqueness of nonnegative solutions and obtain linear stability results. We also give numerical simulations with different values of the small parameter ε and compare the...
We devise a rst-order in time convex splitting scheme for a nonlocal Cahn HilliardOono type equation with a transport term and subject to homogeneous Neu-mann boundary conditions. The presence of the transport term is not a minor modi-cation, since, for instance, we lose the unconditional unique solvability and stability. However, we prove the stab...
Our aim in this paper is to study the well-posedness and the existence of the global attractor of anisotropic Caginalp phase-field type models with singular nonlinear terms. The main difficulty is to prove, in one and two space dimensions, that the order parameter remains in the physically relevant range and this is achieved by deriving proper a pr...
Our aim in this paper is to study the asymptotic behavior, in terms of finite‐dimensional attractors, for higher‐order Navier‐Stokes‐Cahn‐Hilliard systems. Such equations describe the evolution of a mixture of 2 immiscible incompressible fluids. We also give several numerical simulations.
We consider a phase-field system modeling phase transition phenomena, where the Cahn–Hilliard–Oono equation for the order parameter is coupled with the Coleman–Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into...
Our aim in this paper is to study the well-posedness and the dissipativity of higher-order anisotropic conservative phase-field systems. More precisely, we prove the existence and uniqueness of solutions and the existence of the global attractor.
Le glutamate est un neurotransmetteur excitateur majeur, représentant plus de 80 % des neurotransmetteurs dans les flux synaptiques. L’étude des concentrations in vivo en glutamate et glutamine dans le cerveau est cruciale pour comprendre le transport de l’information et ses modifications en condition pathologique.
Malgré l’avènement de nouvelles s...
Our aim in this paper is to study discretized parabolic problems modeling electrostatic micro-electromechanical systems (MEMS). In particular, we prove, both for semi-implicit and implicit semi-discrete schemes, that, under proper assumptions, the solutions are monotonically and pointwise convergent to the minimal solution to the corresponding elli...
This paper investigates the spatial behavior of the solutions of two generalized thermoelastic theories with two temperatures. To be more precise, we focus on the Green–Lindsay theory with two temperatures and the Lord–Shulman theory with two temperatures. We prove that a Phragmén–Lindelöf alternative of exponential type can be obtained in both cas...
Our aim in this article is to review and discuss the Cahn-Hilliard equation, as well as some of its variants. Such variants have applications in, e.g., biology and image inpainting.
We consider the so-called Cahn-Hilliard-Oono equation with singular (e.g. logarithmic) potential in a bounded domain of Rd, d ≤ 3. The equation is subject to an initial condition and Neumann homogeneous boundary conditions for the order parameter as well as for the chemical potential. However, contrary to the Cahn-Hilliard equation, the total mass...
Our aim in this paper is to study properties of a parabolic-elliptic system related with brain lactate kinetics. These equations are obtained from a reaction-diffusion system, when a small parameter vanishes. In particular, we prove the existence and uniqueness of nonnegative solutions and obtain error estimates on the difference of the solutions t...
Our aim in this article is to study the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen-Cahn/Cahn-Hilliard system. In particular, we prove the existence of an exponential attractor and, as a consequence, the existence of the global attractor with finite fractal dimension.
Our aim in this article is to propose a generalization of the Bertozzi– Esedoglu–Gillette–Cahn–Hilliard equation, introduced for binary image inpainting, for grayscale image inpainting. In particular, we consider the solution to the corresponding Cahn–Hilliard inpainting model as a complex valued function. We are interested in the study of the well...
Our aim in this paper is to study higher-order (in space) Allen–Cahn and Cahn–Hilliard
models. In particular, we obtain well-posedness results, as well as the existence of
the global attractor. We also give, for the Allen–Cahn models,
numerical simulations which illustrate the effects
of the higher-order terms and the anisotropy.
Our aim, in this paper, is to study a generalization of the Caginalp phase-field system based on the Gurtin{Pipkin law with two temperatures for heat conduction with memory. In particular, we obtain well-posedness results and study the dissipativity, in terms of the global attractor with optimal regularity, of the associated solution operators. We...
Our aim in this paper is to study higher-order (in space) Allen-Cahn and Cahn-Hilliard models. In particular, we obtain well-posedness results, as well as the existence of the global attractor.
The Caginalp phase-field system has been proposed in [4] as a simple mathematical model for phase transition phenomena. In this paper, we are concerned with a generalization of this system based on the Gurtin-Pipkin law with two temperatures for heat conduction with memory, apt to describe transition phenomena in nonsimple materials. The model cons...
Our aim in this article is to study properties of a reaction-diffusion
system which is related with brain lactate kinetics, when spatial diffusion
is taken into account. In particular, we prove the existence and uniqueness
of nonnegative solutions and obtain linear stability results. We also derive
$L^\infty $-bounds on the solutions. These results...
Our aim in this paper is to study higher-order (in space) anisotropic generalized Cahn-Hilliard models. In particular, we obtain well-posedness results, as well as the existence of the global attractor. Such models can have applications in biology, image processing, etc. We also give numerical simulations which illustrate the effects of the higher-...
Our aim in this paper is to study the well-posedness and the dissipativity of higher-order Cahn–Hilliard equations with dynamic boundary conditions. More precisely, we prove the existence and uniqueness of solutions and the existence of the global attractor.
Our aim in this article is to study the long time behavior, in terms of finite dimensional attractors, of a class of doubly nonlinear Allen–Cahn equations with singular potentials. In particular, we prove the existence of the global attractor which has finite fractal dimension.
Our aim in this paper was to study the well-posedness and the dissipativity of higher-order anisotropic phase-field systems. More precisely, we prove the existence and uniqueness of solutions and the existence of the global attractor.
Our aim in this paper is to study the well-posedness of a singular reaction-diffusion equation which is related with brain lactate kinetics, when spatial diffusion is taken into account. Copyright
Our aim in this chapter was to study higher-order (in space) Allen–Cahn models with logarithmic nonlinear terms. In particular, we obtain well-posedness results, as well as the existence of the global attractor.
The paper concerns with the existence, uniqueness, regularity and the approximation of solutions to the nonlinear phase-field (Allen-Cahn) equa- tion, endowed with non-homogeneous dynamic boundary conditions (depend- ing both on time and space variables). It extends the already studied types of boundary conditions, which makes the problem to be mor...
Projects
Projects (2)
This research thematic is a challenge at many levels. From a mathematical point of view the analysis of fast-slow and delayed systems is a current growing research area. From a medical point of view, brain nutrient mechanisms understanding is a promising problematic. It could help to define suitable therapeutic strategies against tumor or neurodegenerative diseases (less expensive, better adapted, less invasive, faster and with better chances of survival). These projects are also the opportunity to develope mathematical tools to respond to current medical problems.