Research statement L´eo Ag´elas Department of Mathematics, IFP Energies Nouvelles, 1-4, avenue de Bois-Pr´ eau, F-92852 Rueil-Malmaison, France

Keywords: Finite Volume scheme; multiphase Darcy flow in anisotropic heterogeneous porous media; incompressible flows; Finite time singularities;

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My research lies in the area of nonlinear evolution equations. There are many questions which can be asked about evolution equations: the existence of particular types of solutions, such as equilibrium solutions, travelling waves, selfsimilar solutions, time-periodic solutions; the dynamic stability of these solutions; the long time asymptotic behavior of solutions; chaotic dynamics; complete integrability; singular perturbation expressions for solutions; convergence of numerical schemes; and so on. My research covers both the study of numerical methods for ODEs, PDEs equations and the qualitative properties of solutions to both linear and nonlinear elliptic, hyperbolic and parabolic equations, such as regularity, long time behavior, and unique continuation. In particular, I am interested in the mathematical analysis of classical equations of fluid mechanics, such as the Navier- Stokes, the Euler. The solutions of these equations exhibit complex (chaotic or turbulent) behavior, thus rigorous mathematical analysis is crucial for a better understanding of the models and to provide proper guidelines for numerical computations. My research is divided into two parts, the first part is devoted to the discretization of differential operators involved in evolution equations and the second part is devoted to the well-posedness problem of evolution equations. The first part of my research has been made during my thesis and my position as research engineer at IFP Energies nouvelles. The second part of my research has been made during my free-time independently of my work at IFP Energies nouvelles. 1. Discretization of differential operators

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In this part, my research is divided into two stages, the first is devoted to the spatial discretization in particular of low-order of differential operators, hyperbolic and elliptic type and the other devoted to the temporal discretization . In the study of numerical methods for PDEs equations, my research focus on numerical methods for Geoscience modeling of reservoirs, sedimentary basins Email address: [email protected] (L´ eo Ag´ elas)

Preprint

August 30, 2017

and storage : mathematical model formulation, discretization of PDEs by finite volume methods, numerical analysis. 30

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1.1. Spatial discretization of differential operators My research led to the study of: 1.1.1. Finite volume schemes for heterogeneous and anisotropic diffusion problems on general nonconforming meshes Many applications in the oil industry require the simulation of compositional, multiphase Darcy flow in heterogeneous porous media. In the oil industry, the need to improve accuracy of predictions has prompted the use of general unstructured meshes and full permeability tensors. One of the key ingredients for the numerical simulation of Darcy flow in heterogeneous porous media is the discretization of anisotropic heterogeneous elliptic terms such as −div(K∇u). My effort has therefore been devoted to finding consistent and robust Finite Volume discretizations of anisotropic heterogeneous elliptic terms on general meshes. My search was focused on methods which are consistent and coercive on general polyhedral meshes as well as robust with respect to the anisotropy and heterogeneity of the permeability tensor; yield well-conditioned linear systems for which optimal preconditioning strategies can be devised; have a narrow stencil, both to improve matrix sparsity and to reduce the communications in parallel implementations. My quest was motivated in particular by the numerical simulation of complex flows in porous media, which includes coupling with thermodynamics and/or chemistry; because of these complex couplings, the discretization method is often chosen to be cell-centered in industrial codes. To answer to this search, my research led to propose in particular: • In [3, 5], an extension to the Finite volume scheme MPFA O on general polyhedral meshes, this scheme is widely used in the oil industry for the discretization of diffusion fluxes in multiphase Darcy porous media flow model. The well-posedness and convergence of the scheme are derived assuming a local coercivity condition which can be easily checked numerically. This was the first proof of convergence of the MPFA O for general polyhedral meshes as well as L∞ discontinuous diffusion tensor which is essential in many practical applications. • In [7], the finite volume G Scheme , a family of cell-centered finite volume schemes generalizing the MPFA L method, MPFA L method has been proposed by Aavatsmark et al. as an improvement of the MPFA O method both in terms of stencil size and monotonicity properties. We have provided convergence proof covers more general FV schemes expressed in terms of numerical fluxes and it is inspired by the techniques of Eymard, Gallou¨et and Herbin (see, e.g., [50, 51]). The relevant requirements are weak flux consistency and coercivity. To the best of our knowledge, it was the first rigorous convergence proof of the MPFA L method for general meshes and arbitrary heterogeneous anisotropic diffusion tensors. 2

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• Although several schemes were recently proposed, there was yet no ultimate scheme, i.e. a centered scheme with small stencil, which respects the physical bounds and yields good approximations even on non-conforming distorted meshes and with a sharp contrast in the permeabilities. In this way, we proposed Dioptre in [4], a cell-centered symmetric scheme which combines the advantages of the MPFA (multipoint flux approximation) schemes such as the L or the O scheme and of hybrid schemes: it may be used on general non-conforming meshes, it yields a 9-point stencil on two-dimensional quadrangular meshes, it takes into account the heterogeneous diffusion matrix, it is coercive and it can be shown to converge. The scheme relies on the use of special points, called harmonic averaging points, located at the interfaces of heterogeneity. The method used, make an analogy with another domain which is dioptre in optic. • In [6], we propose a unified analysis framework encompassing a wide range of nonconforming discretizations of anisotropic heterogeneous diffusion operators on general meshes. The analysis framework relies on the discrete functional analysis tools for piecewise polynomial spaces, namely the discrete Sobolev-Poincar´e inequality and the discrete Rellich theorem introduced in [50, 51] for finite volume methods and extended in [29, 40] for the discontinuous Galerkin methods. We thus extend the results obtained in [50, 51, 29, 40] by proposing an abstract set of properties ensuring the convergence of a discretization method to minimal regularity solutions. Finite volume schemes as well as the most common discontinuous Galerkin methods are shown to fit in the analysis. • In [12], we prove the convergence of a family of cell-centered non linear Finite Volume scheme. Our proof generalizes the one given in [47] and allows us to obtain the proof of convergence for the nonlinear finite volume schemes introduced in [65, 77, 39, 67, 68, 74] for which no proof yet existed as mentionned in [49]. The main advantage of this family of cell-centered non linear Finite Volume scheme is to produce unconditionally monotone schemes for arbitrary grids and permeability tensors. This property is crucial to have robust approximations of the solution since non-monotone schemes may produce non-physical solutions, which in turn influence the efficiency and convergence of the scheme. Meanwhile, several methods have been recently developed to construct schemes for elliptic PDEs on generic meshes: multipoint flux approximations [1, 2, 7], discontinuous Galerkin [40, 73], discrete duality finite volume [44, 25], mimetic finite difference [28, 20], hybrid finite volume [52], mixed finite volume [46] (it turns out to be identical [48]). However, none of these methods were monotone schemes. 1.1.2. Finite volume schemes for poroelasticity problems on general nonconforming meshes In the industrial world, mechanics-flow coupling is usually ensured by an external coupling of specialized and very rich codes: a code treating multi3

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phase compositional Darcy flows and a code treating mechanic. Such approach is not an easy task, since it involves local remeshing, interpolations between 3D meshes, and external sequential coupling. The alternative solution was to develop a coupled finite volume approach (flow and mechanics are solved simultaneously). Despite a poor literature regarding coupled finite volume approximations of the poroelasticity problem, my research focused on obtaining an unconditionally stable (and symmetric) lowest-order discretization of the quasistatic single-phase poroelasticity problem, which applies on general meshes, and whose convergence can be proved under very low regularity assumptions on the solutions and on the data. For those in [16], we have considered the problem of linear elasticity, it was known that many numerical methods suffer deteriorations in performance as one of the Lam´e constant, λ, goes to infinity. This is the so-called locking phenomenon which arises when the elastic material becomes nearly incompressible. That is, in terms of error estimates, the constant in front of these estimates blows up when the Lam´e constant λ goes to infinity. To overcome the locking, we need to construct numerical methods whose optimal error estimates are uniform with respect to λ ∈]0, +∞[. Many literatures concerning elasticity have appeared to be locking-free. However, to the best of our knowledge,no research work dealt with the locking phenomenon of elasticity with numerical methods for general meshes while keeping the compact stencil (stencil limited to the unknowns of neighbouring cells). Indeed, unstructured, possibly non conforming meshes, including all the degenerate cells obtained from the hexahedron can result from geometries used in reservoir and CO2 storage models to describe complex geometrical features, such as faults, vanishing layers, inclined wells. Furthermore, for cost reasons, numerical methods with compact stencil are wished due to the fact, they are well adapted to industrial codes with parallel architectures. We have introduced a new discretization scheme for linear elasticity equations which should apply on general meshes possibly nonconforming, guarantee to be locking-free and have compact stencil. Moreover, we obtain optimal uniform energy norm estimates for the discretization error. Then, in [17], we have studied the numerical approximation Biot’s consolidation model which is a particular limit case of the poroelasticity problem. 1.1.3. Finite volume method for the simulation of deep-sea sedimentary systems Several petroleum recent discoveries are set in ”turbidite” reservoirs lying at the deep offshore location where they are difficult to study and characterized. These reservoirs are highly heterogeneous and formed by the stacking of deposits related to several hundred turbidity events. One way to better understand their internal architecture of sediment heterogeneity is to understand the processes leading to their formation: the turbidite systems. Turbidity currents are sediment laden underflows that occur in the ocean and in lakes. Turbidity currents are difficult to study in real time and space because of their unpredictable nature and of their size and energy. Then, numerical modelling becomes one way to study these systems. It has the advantage to be easily controlled and scalable to reality in opposition to laboratory experiments. Numerical models are often 4

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used to infer the impact external parameters (such as topography, type and quantity of sediments). In this industrial context, from an approximate layeraveraged equations based on Saint-Venant equations introduced by Parker, to describe the mechanics of turbid underflows, in [19], we have developed a wellbalanced finite volume scheme for hyperbolic systems 1D based on Saint-Venant equations to model the formation of turbidite reservoirs in order to reproduce the distribution of different lithologies such as sand or clay and characterize the architecture deposits at reservoir scale. 1.1.4. Boundary element method for fracture problems in 3D Hydraulic fracturing has been used in the petroleum industry as a stimulation technique to enhance oil and gas recovery in low permeability reservoirs and for estimating in situ stresses. Hydraulic fracturing is a complex operation in which the fluid is pumped at high pressure into a selected section of the wellbore. The injection/extraction operation perturbs the in situ stress state in the reservoir, leading to fracture initiation and or/activation of discontinuities such as faults and joints. These events often manifest as micro seismicity in the reservoir, which can be monitored. To better understand these events, it was necessary to investigate the processes and mechanisms associated with fluid induced changes in the reservoir to improve reservoir development and management. In [18], my research was thus focused on the development of a robust and fast solver based on the 3D elastic Discontinuity Displacement (DD) method in order to study the response of a vast network of fractures in fluid pressure and rock cooling under a given state of stress. 1.2. Temporal discretization

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My research led also to the study of: Numerical methods for Ordinary Differential Equations (ODE): ODEs systems frequently occur as mathematical models in many branches of science, engineering and economy. Because of computational time and memory issues, parallelization has become an indispensable tool in the solution of large scale problems. Thus, my research focuses on numerical methods for solving large-scale nonlinear ODEs systems on parallel computers. The challenge was to find a robust accurate numerical method, and consuming few CPU-time to solve large-scale nonlinear ODEs systems while handling the stiffness of the system. Then, to save CPU-time, we focus on linear multistep methods in order to get good approximations of the differential equation systems while keeping large time step. One of the main difficulties encountered is due to the stiffness of the systems which can lead to numerical instabilities. For all these reasons, we have used BDF methods of order less than or equal to six, the most efficient linear multistep methods to handle stiff equations. Then, in [15], we establish a sharp global error estimate for general autonomous linear ODE Systems using BDF methods. 2. Well-posedness problem of evolution equations

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In this part, my research interests focus on the analysis of parabolic evolution equations. My research lies in the theory of abstract parabolic evolution equations characterized by three main approaches namely, the semigroup methods, the variational methods and the methods of using operational equations (differential, integral, and integro-differential equations). A central issue in the study of nonlinear evolution equations is that solutions may exist locally in time (that is, for short times) but not globally in time. This is caused by a phenomenon called blow-up. The most basic question concerns the existence and uniqueness of solutions. If the evolution equation is to provide a self-consistent mathematical model of some real system then, at a minimum, there should exist a unique solution of this initial value problem. Then, my research focuses on this basic question. Nonlinear equations are usually difficult to analyze. Local existence can be established by standard arguments for most reasonable PDE’s. Global existence is often much harder to establish and typically requires sufficiently strong a priori estimates for solutions of the PDE. The basic global existence and uniqueness questions are not resolved even for some of the most important nonlinear evolution equations arising in classical physics. Then, my research focuses on the analysis nonlinear evolution equations such as: 2.0.1. Geometric non blow-up criteria of Navier-Stokes, Euler and 2D quasigeostrophic equations In this section, my research carry on the 3D Navier-Stokes and Euler equations given by, ( ∂u + (u · ∇)u + ∇p − ν∆u = 0, (1) ∂t ∇ · u = 0,

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in which u = u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t)) ∈ R3 , p = p(x, t) ∈ R and ν ≥ 0 (ν = 0 corresponds to the Euler equations) denote respectively the unknown velocity field, the scalar pressure function of the fluid at the point (x, t) ∈ R3 × [0, ∞[ and the viscosity of the fluid. My research carry also on the the 2D Quasi-Geostrophic equation (2) in R2 given by   ∂u + v · ∇u = 0, (2) ∂t  v = −∇⊥ (−∆)− 12 u,

here ∇⊥ = (−∂x2 , ∂x1 ). One of the most challenging questions in fluid dynamics is whether the three- dimensional (3D) incompressible Navier-Stokes, Euler and two-dimensional QuasiGeostrophic (2D QG) equations can develop a finite-time singularity from smooth initial data. In two recent papers [75, 76], it has been shown a significant barrier to establishing global regularity for the three-dimensional Euler, in that any method for achieving this must use the finer geometric structure of these equations. Indeed, the development of finite-time singularities was first investigated via energy methods which focus on global features or on point-wise Eulerian features of the flow. This comes at the disadvantage of neglecting 6

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the structures and physical mechanisms of the flow evolution. A strategy to overcome such shortcomings was established by focusing more on geometrical properties and flow structures, such as vortex tubes or vortex lines (see [32, 21, 22, 36, 33, 38, 35, 41, 43]). Furthermore, the results obtained in [76] indicate that translation and rotation invariance present in the Navier-Stokes 3D Euler and 2D QG equations and characterized by the Biot-Savart law (see [76]), may be the key for the non blow-up in finite time of the solutions. Then in [14], we establish a new geometric criterion for non blow-up in finite time of the solutions of Navier, Euler and 2D QG equations based on the regularity of the direction of the vorticity ξ in regions containing the positions where the maximum of the magnitude of the vorticity is reached and shrinking to zero as time tends to some T ∗ the alleged time of singularity. We thus obtain a regularity criterion which appears less restrictive than the Beale-KatoMadja (BKM) regularity criterion type. Furthermore, due to the existence of hyperbolic-saddle singularities suggested by the generation of strong fronts in geophysical/meteorology observations (see [35, 37]), and antiparallel vortex line pairing observed in numerical simulations and physical experiments, it was important to take them into account in our geometric non blow-up criterion, which is the case. We establish also the non blow-up in finite time of the solutions of NavierStokes, 3D Euler and 2D QG equations by considering the most plausible blowup rates for kω(t)k∞ , ku(t)k∞ and k∇ξ(t)k∞ . To obtain this result, we overcomed the obstruction that we do not know if there exists an isolated absolute maximum for the vorticity achieved along a smooth curve as it was assumed in Proposition 2.1 of [35]. Moreover, recent numerical experiments show that it is not always the case (see [62], see also section 5.4.5 in [56]). We overcome this difficulty by using a result of Pshenichnyi concerning directional derivatives of the function of maximum and the structure of a set of supporting functionals [70]. 2.0.2. Regularity issue of Magneto-hydrodynamics equations In this subsection my research carry on the initial-value problem for the 2D incompressible magneto-hydrodynamics equations with Laplacian magnetic diffusion,  = −∇p + (b · ∇)b  ∂t u + (u · ∇)u ∂t b + (u · ∇)b−∆b = (b · ∇)u (3)  ∇ · u = 0, ∇ · b = 0 with initial conditions

u(x, 0) = u0 (x) for a.e x ∈ R2 b(x, 0) = b0 (x) for a.e x ∈ R2 . 260

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Magneto-hydrodynamics equations (MHD) describe the evolution of an electrically conducting fluid such as plasma (A plasma is a hot, ionized gas containing free electrons and ions), liquid metal (for example, mercury or liquid sodium), and salt water or electrolyte and thus, MHD equations are expressed as the 7

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combinations of the fluid equations and Maxwell system. The fundamental concept behind MHD is that magnetic fields can induce currents in a moving conductive fluid, which in turn creates forces on the fluid and also changes the magnetic field itself. There are two serious technological applications of MHD that may become very important in the future. In the first, strong magnetic fields are used to confine rings or columns of hot plasma that will be held in place long enough for thermonuclear fusion to occur and for net power to be generated. In the second, which is directed toward a similar goal, liquid metals are driven through a magnetic field in order to generate electricity. The study of magnetohydrodynamics is also motivated by its widespread application to the description of space (within the solar system) and astrophysical plasmas (beyond the solar system). Due to their prominent roles in modeling many phenomena in astrophysics, geophysics and plasma physics, the MHD equations have been studied extensively mathematically. Thus, besides its physical applications , the MHD equations is also mathematically significant. For the two-dimensional incompressible MHD equations with positive fluid viscosity and magnetic diffusivity, it is well known that there exists a unique global classical solution for every initial data. But if the viscosity of the fluid or the magnetic diffusivity is zero, then the global regularity issue for the 2D MHD equations remains as an open problem. In the case that the viscosity is zero and the magnetic diffusivity is positive, in [63, 30, 64], the authors obtained the global existence of a more regular weak solution. But still the global existence of smooth solutions remains as a challenging open problem. The main reason for the unavailability of a proof of global regularity is due to the quadratic coupling between the velocity field and the magnetic field which invalidates the vorticity conservation. In [9], my research on this topic lead to a logarithmically supercritical improvement of the previous results obtained in [31, 59]. More precisely by considering the two-dimensional MHD equations with only the magnetic diffusion given by D a Fourier multiplier whose symbol m is given by m(ξ) = |ξ|2 log(e+|ξ|2 )β , we prove that there exists an unique global solution in H s (R2 ) with s > 2 for these equations when β > 1. This result improves the previous works [31, 59] which require that m(ξ) = |ξ|2β with β > 1 for which it was no longer possible to proceed as [59] and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely m(ξ) = |ξ|2 . In [11], we derive a regularity criterion less restrictive than the Beale-KatoMadja regularity criterion type, namely any solution (u, b) ∈ C([0, T [; H r (R2 )) with r > 2 remains in H r (R2 ) up to time T under the assumption that 1 Z T 2 k∇u(t)k∞ dt < +∞. This regularity criterion may stand as a 0 log(e + k∇u(t)k∞ ) great improvement over the usual BKM regularity criterion which states that Z T if k∇ × u(t)k∞ dt < +∞ then the solution (u, b) ∈ C([0, T [; H r (R2 )) with 0

r > 2 remains in H r (R2 ) up to time T . Indeed, in virtue of ∇u = R((∇ × u) Id) 8

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with R Riesz transform on matrix-valued functions, we expect that the blow-up rate at a time T of k∇u(t)k∞ be the same as the one of k∇ × u(t)k∞ and due to the quadratic nonlinearity of the MHD equations, be at least faster than  1 O . Moreover, our result applies also to a class of equations arising in T −t hydrodynamics and studied in [53] for their L∞ ill-posedness. 2.0.3. Regularity issue of the conserved Kuramoto-Sivashinsky (cKS) equation In this subsection, my research carries on the conserved Kuramoto-Sivashinsky (cKS) equation described by the following partial differential equation, ∂t v + ∆2 v + ∆|∇v|2 = 0

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v(0) = v0

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with initial condition

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on Rd with solutions vanishing at infinity as |x| → ∞ or on the d−dimensional torus Td ≡ Rd /(2πZ)d , d ∈ N∗ with periodic boundary conditions and in this case we require in addition that R v0 is a periodic scalar function of period 2π with zero mean value, that is Td v0 (x) dx = 0. The cKS equation models the step meandering instability on a surface characterized by the alternation of terraces with different properties [54]. It appeared as a model for the boundaries of terraces in the epitaxy of Silicon [54]. It also describes the growth of an amorphous thin film by physical vapor deposition [71] and [72] (some applications requiring thin films include solar cells, mechanical coatings, and, more recently, microelectromechanical systems and microfluidic devices). Growth conditions have a profound effect on the morphological quality of films [78] and has recently received increasing interest in materials science. One of the outstanding challenges is to understand these growth processes qualitatively and quantitatively, so that control laws can be formulated which optimize certain film properties, e.g., flatness, conductivity. In consequence, the mathematical models for the study of surface growth and the experiments done thus improving these models in terms of physics has attracted a lot of attention in recent years. However, up today, existence and uniqueness theorem of global strong solutions of the fourth order parabolic equations modeling epitaxy thin film growth remains a challenging open problem in the two-dimensional case and even in the one-dimensional case [23, 24]. The main difficulties in treating such equations are caused by the nonlinearity terms and the lack of a maximum principle. In [8], we show by taking into account of the main physical phenomena and a better approximation of terms related to them in the mathematical model, lead to a kind of cancellation of nonlinear terms between them in some spaces and from this, we obtain existence and uniqueness of global strong solutions for such equations in Rd , d ∈ {1, 2}. In [10], similarly as in [66], we partially answer to the existence and smoothness problem for cKS equations by showing the finite time blow up of complexvalued solutions associated to a class of large initial data, more precisely, we 9

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show that there is complex-valued initial data that exists in every Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the complex-valued solution is in no Besov space (and hence in no Lebesgue or Sobolev space). 2.0.4. Regularity issue of 2D Quantum Navier-Stokes equations In this subsection, my research carry on the Cauchy problem for Quantum Navier Stokes equations as follows:  ∂n   + ∇ · (nu) = 0, ∂t  √  (7) ∂nu ∆ n 2   − 2ε∇ · (nDu) + ∇p(n) = 0 + ∇ · (nu ⊗ u) − 2ε n∇ √ ∂t n with initial conditions,

n(x, 0) = n0 , u(x, 0) = u0 ,

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on Ω = R2 /LZ2 , L > 0 with periodic boundary conditions and we require in adT dition that n and u are periodic functions of period L where Du = ∇u + ∇u , 0

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p(n) = gn2 with g > 0 and ε > 0 is the scaled Planck constant. From [58, 61], it is assumed that the scaled Planck constant ε is of order 10−2 .

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Quantum hydrodynamic models become important and necessary to model and simulate electron transport, affected by extremely high electric fields, in ultra-small sub-micron semiconductor devices, such as resonant tunnelling diodes, where quantum effects (like particle tunnelling through potential barriers and built-up in quantum wells take place and dominate the process. They arise in semiclassical mechanics in the study of semiconductor devices. Quantum fluid models are used to describe, superfluids, quantum semiconductors, weakly interacting Bose gases and quantum trajectories of Bohemian mechanics. Quantum hydrodynamics have engendered substantial activity in the field of theoretical chemical dynamics. In quantum chemistry, they arise as solutions to chemical kinetic systems, in which case they are derived from the Schr¨odinger equation by way of Madelung equations [69]. However, up today, global existence and uniqueness theorem of strong solutions for the 2D Quantum Navier-Stokes remains a challenging open problem. There are only few mathematical results for these viscous quantum hydrodynamic models due to difficulties coming from the third order derivatives. The existence of global-in-time weak solutions in one space dimension without smallness conditions is proved in [55]. Concerning the multidimensional case, local-in-time existence theorems have been obtained in [34, 45]. Global-in-time existence of weak solutions in a three-dimensional torus for large data is achieved in [60]. This proof of existence relies on the reformulation of the Quantum Navier-Stokes model as a viscous quantum Euler system by using the idea developed for the Bresch-Desjardin Entropy in [26, 27].

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In [13], we prove existence and uniqueness of global strong solutions of Quantum Navier-Stokes equations in R2 for any initial data (n0 , u0 ) with n0 a small perturbation of some positive constant n0 > 0 and such that n0 is sufficiently large relative to (n0 −n0 , u0 ) in some sense. To our knowledge, this result is new and gives a positive answer to the open problem of existence and smoothness of global solutions of such equations. References

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[1] Aavatsmark, I.,Barkve, T., Boe, O. and Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods, SIAM J. Sci. Comput., 19 (1998), pp. 1700-1716. [2] Aavatsmark, I., Barkve, T., Boe, O. and Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part II: Discussion and numerical results, SIAM J. Sci. Comput., 19 (1998), pp. 1717-1736.

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[3] Ag´elas, L. and Masson, R.: Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes. C. R. Acad. Sci. Paris. S´er 1. (346) 1007-1012. October 2008. [4] Ag´elas, L., Eymard, R. and Herbin, R.: A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. C. R. Acad. Sci. Paris. S´er 1. (347) 673-676. June 2009. [5] Ag´elas, L., Guichard, C. and Masson, R.: Convergence of Finite Volume MPFA O type Schemes for Heterogeneous Anisotropic Diffusion Problems on General Meshes. International Journal on Finite Volumes 7, 2 (2010), 1-33.

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[6] Ag´elas, L., Di Pietro, D. A., Eymard, R. and Masson, R. : An abstract analysis framework for nonconforming approximations of the single phase Darcy equation,, International Journal on Finite Volumes 7, 1 (2010), 1-29. [7] Ag´elas, L., Di Pietro, D. and Droniou, J.: The G method for heterogeneous anisotropic diffusion on general meshes. M2AN Math. Model. Numer. Anal. 44 (2010). No. 4, 597-625. [8] Ag´elas, L.: Global regularity of solutions of equation modeling epitaxy thin film growth. Journal of Evolution Equations, vol. 15, Issue. 1, pp. 89-106, March 2015.

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[9] Ag´elas, L.: Global regularity for a logarithmically critical 2D MHD equations with zero viscosity. Monatshefte f¨ ur Mathematik, 181 (2016). No 2, 245-266. [10] Ag´elas, L.: Finite time blow up of complex solutions of the conserved Kuramoto-Sivashinsky equation in Rd , d ≥ 1. Accepted for publication in Proceedings of the Royal Society of Edinburgh. Section A. 11

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[11] Ag´elas, L.: Beyond the BKM’s criterion for the 2D resistive magnetohydrodynamic equations. To be appeared. [12] Ag´elas, L., Enchery, G., Flemischb, B., Schneider, M.: Convergence of nonlinear finite volume schemes for heterogeneous anisotropic diffusion on general meshes. To be appeared.

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[13] Ag´elas, L.: Global regularity of Quantum Navier-Stokes equations in R2 . Submitted. [14] Ag´elas, L.: Some advances on the geometric non blow-up criteria of incompressible flows. Submitted.

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[15] Ag´elas, L, Gallois, T.-H.: A sharp error estimate for linear ODE Systems. Submitted. [16] Ag´elas, L and Lemaire, S.: A locking-free compact stencil finite volume scheme for linear elasticity on general meshes. Ongoing paper.

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[17] Ag´elas, L., Eymard, R. and Lemaire, S.: Convergence of Euler-Gradient approximations for Biot’s consolidation problem on general meshes. Ongoing paper. [18] Ag´elas, L., Baroni, A., Ben Gharbia, I., Colombo, D. and Schueller, S.: A fast Displacement Discontinuity Solver for elastostatic fracture problems in 3D. Ongoing paper.

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[19] Benarbia, Mohamed Amine: Master Mod´elisation et Simulation. University of Versailles- Saint-Quentin-en-Yvelines,France. Co-supervised with Vanessa Teles, Research Engineer in Geology. Thesis title: M´ethodes num´eriques pour la simulation des syst`emes s´edimentaires marins profonds. [20] Beirao da Veiga, L. and Manzini, G.: A higher-order formulation of the mimetic finite difference method, SIAM J. Sci. Comput., 31 (2008), pp. 732-760. [21] Beir˜ ao da Veiga, H. and Berselli, L. C.: Navier-Stokes equations: Green’s matrices, vorticity direction, and regularity up to the boundary, J. Differ. Equations, 246, 597-628, (2009).

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[22] Berselli, L.C.: Some geometric constraints and the problem of global regularity for the Navier-Stokes equations, Nonlinearity, 22(10),2561-2581, (2009). [23] Bl¨ omker, D., Romito, M. : Local existence and uniqueness in the largest critical space for a surface growth model, Nonlinear Differ. Equ. Appl. 19 (2012), 365-381.

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