Alain Miranville

Université Le Havre Normandie

See https://my.siam.org/Store/Product/viewproduct/?ProductId=30927347 and https://epubs.siam.org/doi/book/10.1137/1.9781611975925

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...

Protein design with desirable properties has been a significant challenge for many decades. Generative artificial intelligence is a promising approach and has achieved great success in various protein generation tasks. Notably, diffusion models stand out for their robust mathematical foundations and impressive generative capabilities, offering uniq...

Our aim in this paper is to study the regularity of solutions to a variant of the Cahn-Hilliard equation with logarithmic nonlinear terms. This model was proposed by S. Forest and is based on microconcentrations. It has applications to Lithium-Ion batteries and is efficient in numerical simulations. Furthermore, we are able to prove the strict sepa...

In this paper, we consider the asymptotic behavior of weak solutions for nonclassical non-autonomous diffusion equations with a delay operator in time-dependent spaces when the nonlinear function $g$ satisfies subcritical exponent growth conditions, the delay operator $\varphi(t, u_t)$ contains some hereditary characteristics and the external force...

The structure stability from the Gromov-Hausdorff viewpoint can be used to describe the continuity of complete trajectories inside global attractors for nonlinear evolutionary partial differential equations defined on perturbed domains, such as the unsteady incompressible fluid flow models. This paper is concerned with the Gromov-Hausdorff stabilit...

This paper is concerned with the determination of the 3D vector-valued diffusive Burgers equation in a periodic domain. Taking the initial data in H 1/2 or H 1 , the existence of determining modes via a defined time-dependent determined wavenumbers is established, together with estimates on the time average for the determining wavenumbers.

The present paper is concerned with a fourth‐order Allen–Cahn model with logarithmic potential and mass source that describes the process of phase separation in two‐component systems accompanied by a flux of material. The existence of a global weak solution is obtained under appropriate hypotheses on the source term. Furthermore, we study its Cahn–...

A qualitative study for a second-order boundary value problem with local or nonlocal diffusion and a cubic nonlinear reaction term, endowed with in-homogeneous Cauchy–Neumann (Robin) boundary conditions, is addressed in the present paper. Provided that the initial data meet appropriate regularity conditions, the existence of solutions to the nonloc...

In this paper, we focus on the averaging principle for a class of stochastic differential equations driven by a small fractional Brownian noise. The fast component is driven by a Brownian motion, while the slow component is driven by a fractional Brownian motion with Hurst parameter H (1 3 < H ≤ 1 2). We use the Poisson equation approach to derive...

The aim of this paper is to construct invariant manifolds for a coupled system, consisting of a parabolic equation and a second-order ordinary differential equation, set on T3 and subject to periodic boundary conditions. Notably, the “spectral gap condition" does not hold for the system under consideration, leading to the use of the spatial averagi...

Our aim in this paper is to study a perturbation of the Cahn-Hilliard equation with nonlinear terms of logarithmic type. This new model is based on an unconstrained theory recently proposed in [5]. We prove the existence, regularity and uniqueness of solutions, as well as (strong) separation properties of the solutions from the pure states, also in...

In this study, we introduced a mathematical model mimicking as much as possible the evolutions and interactions between glioma and lactate in the brain, in order to test different therapies and administration protocols. We simulated both glioma cell density evolution and lactate concentration, and considered two therapies: chemotherapy and a treatm...

We consider the hyperbolic relaxation of the viscous Cahn–Hilliard equation with a symport term. This equation is characterized by the presence of the additional inertial term τDϕtt$$ {\tau}_D{\phi}_{tt} $$ that accounts for the relaxation of the diffusion flux. We suppose that τD$$ {\tau}_D $$ is dominated by the viscosity coefficient δ$$ \delta $...

Glial tumors represent the leading etiology of primary brain tumors. Their particularities lie in (i) their location in a highly functional organ that is difficult to access surgically, including for biopsy, and (ii) their rapid, anisotropic mode of extension, notably via the fiber bundles of the white matter, which further limits the possibilities...

This paper is concerned with the Gromov–Hausdorff stability of global attractors for the 3D Navier–Stokes equations with damping under variations of the domain, which describes the complexity of the dynamics of the motion of a fluid flow. The Gromov–Hausdorff stability accounts for the Gromov–Hausdorff distance between two global attractors which m...

The present paper is concerned with a fourth order Allen-Cahn model with mass source, which describes the process of phase separation in two component systems accompanied by flux of material. The existence of global weak solution is obtained under appropriate hypothesis on source term, which leads the homologous Cahn-Hilliard limit as the small par...

In this paper, we consider the asymptotic behavior of weak solutions for non-autonomous diffusion equations with delay in time-dependent spaces when the nonlinear function $f$ is critical growth, the delay term $g(t, u_t)$ contains some hereditary characteristics and the external force $h \in L_{l o c}^{2}\left(\mathbb{R} ; L^{2}(\Omega)\right)$. F...

In this paper, the effect of inhibition of monocarboxylate transporters on intracellular and capillary lactate concentrations is investigated using an optimal control problem. A control term representing the concentration of the inhibitor is used in an ODE model that models lactate kinetics between the cell and the capillary. Finally, some numerica...

Despite impressive empirical advances of SSL in solving various tasks, the problem of understanding and characterizing SSL representations learned from input data remains relatively under-explored. We provide a comparative analysis of how the representations produced by SSL models differ when masking parts of the input. Specifically, we considered...

This paper is concerned with the dynamics of the two-dimensional Navier–Stokes equations with multi-delays in a Lipschitz-like domain, subject to inhomogeneous Dirichlet boundary conditions. The regularity of global solutions and of pullback attractors, based on tempered universes, is established, extending the results of Yang, Wang, Yan and Miranv...

Our aim in this paper is to study a Cahn-Hilliard type system based on microconcentrations. We prove the existence and uniqueness of solutions to this system and then prove the convergence of the solutions to those of the original Cahn-Hilliard equation as a small parameter goes to zero, on finite time intervals.

In this paper, we study the strong global attractors for a three dimensional nonclassical diffusion equation with memory. First, we prove the existence and uniqueness of strong solutions for the equations by the Galerkin method. Then we prove the existence of global attractors for the equations in $H^2(\Omega)\cap H^1_0(\Omega)\times L^2_\mu(\mathb...

This paper is concerned with the existence and regularity of global attractor $\mathcal A$ for a Kirchhoff wave equation with strong damping and memory in the weighted time-dependent spaces $\mathcal H$ and $\mathcal H^{1}$, respectively. In order to obtain the existence of $\mathcal A$, we mainly use the energy method in the priori estimations, an...

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...

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 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 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.

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 + .

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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...