Alexis Vasseur | The University of Texas at Austin (original) (raw)
Papers by Alexis Vasseur
Journal of Computational Physics, 2010
We present a generalized discontinuous Galerkin method for a multicomponent compressible barotrop... more We present a generalized discontinuous Galerkin method for a multicomponent compressible barotropic Navier-Stokes system of equations.
We prove the global existence and uniqueness of strong solutions for a compressible multifluid de... more We prove the global existence and uniqueness of strong solutions for a compressible multifluid described by the barotropic Navier-Stokes equations in dim = 1. The result holds when the diffusion coefficient depends on the pressure. It relies on a global control in time of the L 2 norm of the space derivative of the density, via a new kind of entropy.
We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equati... more We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equations which arise in chemical dynamics by implementing a mixed/discontinuous Galerkin (MDG) finite element numerical scheme. We show that this methodology is stable, showing good accuracy and a remarkable scale invariance in its solution space. In addition the MDG method is robust, adapting well to various initial-boundary value problems of particular significance in a range of physical and chemical applications. We further show explicitly how to recover the Lagrangian frame (or pathline) solutions.
Journal of Computational Physics, 2009
We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equati... more We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equations which arise in chemical dynamics by implementing a mixed/discontinuous Galerkin (MDG) finite element numerical scheme. We show that this methodology is stable, showing good accuracy and a remarkable scale invariance in its solution space. In addition the MDG method is robust, adapting well to various initial-boundary value problems of particular significance in a range of physical and chemical applications. We further show explicitly how to recover the Lagrangian frame (or pathline) solutions.
The aim of this article is to study the existence and uniqueness of strong solutions for a class ... more The aim of this article is to study the existence and uniqueness of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions. The discussions are based on analytic semigroup theory and fixed point theorems. An example illustrates the main results.
We obtain an entropy functional for the Lifshitz-Slyozov system. This can be used to investigate ... more We obtain an entropy functional for the Lifshitz-Slyozov system. This can be used to investigate the time asymptotics of the system. In particular, we describe situations in which the monomers concentration either tend to 0 or saturate as time becomes large. The latter situation can be excluded under assumptions on the support of the initial data.
Asymptotic Analysis
We consider in this paper a plasma subject to a strong deterministic magnetic field and we invest... more We consider in this paper a plasma subject to a strong deterministic magnetic field and we investigate the effect on this plasma of a stochastic electric field. We show that the limit behavior, which corresponds to the transfer of energy from the electric wave to the particles (Landau phenomena), is described by a Spherical Harmonics Expansion (SHE) model.
We consider the compressible Navier-Stokes equation with density dependent viscosity coefficients... more We consider the compressible Navier-Stokes equation with density dependent viscosity coefficients, focusing on the case where those coefficients vanish on vacuum. We prove the stability of weak solutions both in the torus and in the whole space in dimension 2 and 3. The pressure is given by p=rho^gamma, and our result holds for any gamma>1. In particular, we obtain the
Nonlinear Partial Differential Equations, 2011
ABSTRACT In 1957, E. De Giorgi [7] solved the 19th Hilbert problem by proving the regularity and ... more ABSTRACT In 1957, E. De Giorgi [7] solved the 19th Hilbert problem by proving the regularity and analyticity of variational (“energy minimizing weak”) solutions to nonlinear elliptic variational problems. In so doing, he developed a very geometric, basic method to deduce boundedness and regularity of solutions to a priori very discontinuous problems. The essence of his method has found applications in homogenization, phase transition, inverse problems, etc.
In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for regularity of solutions... more In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for regularity of solutions for the Navier-Stokes equation in three dimensions which incorporates weak LpL^pLp norms in the space variables and log improvement in the time variable.
Mathematics of Computation, 2001
We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, th... more We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.
SIAM Journal on Mathematical Analysis, 2008
We consider the Navier-Stokes equations for compressible viscous flu- ids in one dimension. It is... more We consider the Navier-Stokes equations for compressible viscous flu- ids in one dimension. It is a well known fact that if the initial data are smooth and the initial density is bounded by below by a positive constant, then a strong solution exists for a small time. In this paper, we show that under the same hypothesis, the density remains
SIAM Journal on Applied Mathematics, 2002
... LIFSHITZ-SLYOZOV LIMIT* JEAN-FRANCOIS COLLET*, THIERRY GOUDONt, FRÉDÉRIC POUPAUDt, AND ALEXIS... more ... LIFSHITZ-SLYOZOV LIMIT* JEAN-FRANCOIS COLLET*, THIERRY GOUDONt, FRÉDÉRIC POUPAUDt, AND ALEXIS VASSEURt Abstract. We investigate the connection between two classical models for the study of phase transition phenomena, the Becker-Döring equations ...
Monatshefte für Mathematik, 2009
We consider the full system of compressible Navier-Stokes equations for heat conducting fluid. We... more We consider the full system of compressible Navier-Stokes equations for heat conducting fluid. We show that the temperature is uniformly positive for t ≥ t 0 (for any t 0 > 0) for any solutions with finite initial entropy. The assumptions on the viscosity and conductivity coefficients are minimal (for instance, the solutions constructed by E. Feireisl in [2] verify all the requirements).
Journal of Statistical Physics, 2009
This article is devoted to the proof of the hydrodynamic limit for a discrete velocity Boltzmann ... more This article is devoted to the proof of the hydrodynamic limit for a discrete velocity Boltzmann equation before appearance of shocks in the limit system.
Journal of Mathematical Fluid Mechanics, 2011
In this article we present a Ladyženskaja-Prodi-Serrin Criteria for regularity of solutions for t... more In this article we present a Ladyženskaja-Prodi-Serrin Criteria for regularity of solutions for the Navier-Stokes equation in three dimensions which incorporates weak L p norms in the space variables and log improvement in the time variable. . Primary 76D03; Secondary 76D05, 35Q30. This equation is complemented by an initial condition u(0) = u 0 and some condition for the behavior as |x| → ∞, typically |u| → 0 which is made precise by considering solutions in certain functions spaces. The literature regarding solutions for these equations is quite large and we discuss only a small subset which is immediately relevant to this paper. A general open question for solutions is to discover conditions which guarantee a solution is "regular" (or smooth) for all time. For example, given an initial condition u 0 ∈ L 2 (R 3 ) Leray [12] proved there exists a solution (typically called a Leray-Hopf solution, see also ) u ∈ L ∞ ((0, ∞); L 2 (R 3 )) and ∇u ∈ L 2 ((0, ∞); L 2 (R 3 )). If u 0 is regular enough, the solution immediately enters the class of C ∞ smooth functions and remains there for some possibly small time (i.e. u ∈ C ∞ ((0, T * ) × R 3 ) but it remains an open question weather it retains this smoothness property for all time (i.e. u ∈ C ∞ ((0, ∞) × R 3 ).
Journal of Mathematical Analysis and Applications, 2009
We consider the evolution of a quantity advected by a compressible flow and subject to diffusion.... more We consider the evolution of a quantity advected by a compressible flow and subject to diffusion. When this quantity is scalar it can be, for instance, the temperature of the flow or the concentration of some pollutants. Because of the diffusion term, one expects the equations to have a regularizing effect. However, in their Euler form, the equations describe the evolution of the quantity multiplied by the density of the flow. The parabolic structure is thus degenerate near vacuum (when the density vanishes). In this paper we show that we can nevertheless derive uniform L p bounds that do not depend on the density (in particular the bounds do not degenerate near vacuum). Furthermore the result holds even when the density is only a measure.
Journal of Hyperbolic Differential Equations, 2009
We study a mathematical model for sprays which takes into account particle break-up due to drag f... more We study a mathematical model for sprays which takes into account particle break-up due to drag forces. In particular, we establish the existence of global weak solutions to a system of incompressible Navier-Stokes equations coupled with a Boltzmann-like kinetic equation. We assume the particles initially have bounded radii and bounded velocities relative to the gas, and we show that those bounds remain as the system evolves. One interesting feature of the model is the apparent accumulation of particles with arbitrarily small radii. As a result, there can be no nontrivial hydrodynamical equilibrium for this system.
Journal of Differential Equations, 2009
The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynami... more The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations is proved. The model consists of the conservation laws for the particle density and particle current density, including quantum corrections from the Bohm potential and viscous stabilizations arising from quantum Fokker-Planck interaction terms in the Wigner equation. The model equations are coupled self-consistently to the Poisson equation for the electric potential and are supplemented with periodic boundary and initial conditions. When a diffusion term linearly proportional to the velocity is introduced in the momentum equation, the positivity of the particle density is proved. This term, which introduces a strong regularizing effect, may be viewed as a classical conservative friction term due to particle interactions with the background temperature. Without this regularizing viscous term, only the nonnegativity of the density can be shown. The existence proof relies on the Faedo-Galerkin method together with a priori estimates from the energy functional.
Journal of Computational Physics, 2010
We present a generalized discontinuous Galerkin method for a multicomponent compressible barotrop... more We present a generalized discontinuous Galerkin method for a multicomponent compressible barotropic Navier-Stokes system of equations.
Journal of Computational Physics, 2010
We present a generalized discontinuous Galerkin method for a multicomponent compressible barotrop... more We present a generalized discontinuous Galerkin method for a multicomponent compressible barotropic Navier-Stokes system of equations.
We prove the global existence and uniqueness of strong solutions for a compressible multifluid de... more We prove the global existence and uniqueness of strong solutions for a compressible multifluid described by the barotropic Navier-Stokes equations in dim = 1. The result holds when the diffusion coefficient depends on the pressure. It relies on a global control in time of the L 2 norm of the space derivative of the density, via a new kind of entropy.
We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equati... more We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equations which arise in chemical dynamics by implementing a mixed/discontinuous Galerkin (MDG) finite element numerical scheme. We show that this methodology is stable, showing good accuracy and a remarkable scale invariance in its solution space. In addition the MDG method is robust, adapting well to various initial-boundary value problems of particular significance in a range of physical and chemical applications. We further show explicitly how to recover the Lagrangian frame (or pathline) solutions.
Journal of Computational Physics, 2009
We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equati... more We present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equations which arise in chemical dynamics by implementing a mixed/discontinuous Galerkin (MDG) finite element numerical scheme. We show that this methodology is stable, showing good accuracy and a remarkable scale invariance in its solution space. In addition the MDG method is robust, adapting well to various initial-boundary value problems of particular significance in a range of physical and chemical applications. We further show explicitly how to recover the Lagrangian frame (or pathline) solutions.
The aim of this article is to study the existence and uniqueness of strong solutions for a class ... more The aim of this article is to study the existence and uniqueness of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions. The discussions are based on analytic semigroup theory and fixed point theorems. An example illustrates the main results.
We obtain an entropy functional for the Lifshitz-Slyozov system. This can be used to investigate ... more We obtain an entropy functional for the Lifshitz-Slyozov system. This can be used to investigate the time asymptotics of the system. In particular, we describe situations in which the monomers concentration either tend to 0 or saturate as time becomes large. The latter situation can be excluded under assumptions on the support of the initial data.
Asymptotic Analysis
We consider in this paper a plasma subject to a strong deterministic magnetic field and we invest... more We consider in this paper a plasma subject to a strong deterministic magnetic field and we investigate the effect on this plasma of a stochastic electric field. We show that the limit behavior, which corresponds to the transfer of energy from the electric wave to the particles (Landau phenomena), is described by a Spherical Harmonics Expansion (SHE) model.
We consider the compressible Navier-Stokes equation with density dependent viscosity coefficients... more We consider the compressible Navier-Stokes equation with density dependent viscosity coefficients, focusing on the case where those coefficients vanish on vacuum. We prove the stability of weak solutions both in the torus and in the whole space in dimension 2 and 3. The pressure is given by p=rho^gamma, and our result holds for any gamma>1. In particular, we obtain the
Nonlinear Partial Differential Equations, 2011
ABSTRACT In 1957, E. De Giorgi [7] solved the 19th Hilbert problem by proving the regularity and ... more ABSTRACT In 1957, E. De Giorgi [7] solved the 19th Hilbert problem by proving the regularity and analyticity of variational (“energy minimizing weak”) solutions to nonlinear elliptic variational problems. In so doing, he developed a very geometric, basic method to deduce boundedness and regularity of solutions to a priori very discontinuous problems. The essence of his method has found applications in homogenization, phase transition, inverse problems, etc.
In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for regularity of solutions... more In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for regularity of solutions for the Navier-Stokes equation in three dimensions which incorporates weak LpL^pLp norms in the space variables and log improvement in the time variable.
Mathematics of Computation, 2001
We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, th... more We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.
SIAM Journal on Mathematical Analysis, 2008
We consider the Navier-Stokes equations for compressible viscous flu- ids in one dimension. It is... more We consider the Navier-Stokes equations for compressible viscous flu- ids in one dimension. It is a well known fact that if the initial data are smooth and the initial density is bounded by below by a positive constant, then a strong solution exists for a small time. In this paper, we show that under the same hypothesis, the density remains
SIAM Journal on Applied Mathematics, 2002
... LIFSHITZ-SLYOZOV LIMIT* JEAN-FRANCOIS COLLET*, THIERRY GOUDONt, FRÉDÉRIC POUPAUDt, AND ALEXIS... more ... LIFSHITZ-SLYOZOV LIMIT* JEAN-FRANCOIS COLLET*, THIERRY GOUDONt, FRÉDÉRIC POUPAUDt, AND ALEXIS VASSEURt Abstract. We investigate the connection between two classical models for the study of phase transition phenomena, the Becker-Döring equations ...
Monatshefte für Mathematik, 2009
We consider the full system of compressible Navier-Stokes equations for heat conducting fluid. We... more We consider the full system of compressible Navier-Stokes equations for heat conducting fluid. We show that the temperature is uniformly positive for t ≥ t 0 (for any t 0 > 0) for any solutions with finite initial entropy. The assumptions on the viscosity and conductivity coefficients are minimal (for instance, the solutions constructed by E. Feireisl in [2] verify all the requirements).
Journal of Statistical Physics, 2009
This article is devoted to the proof of the hydrodynamic limit for a discrete velocity Boltzmann ... more This article is devoted to the proof of the hydrodynamic limit for a discrete velocity Boltzmann equation before appearance of shocks in the limit system.
Journal of Mathematical Fluid Mechanics, 2011
In this article we present a Ladyženskaja-Prodi-Serrin Criteria for regularity of solutions for t... more In this article we present a Ladyženskaja-Prodi-Serrin Criteria for regularity of solutions for the Navier-Stokes equation in three dimensions which incorporates weak L p norms in the space variables and log improvement in the time variable. . Primary 76D03; Secondary 76D05, 35Q30. This equation is complemented by an initial condition u(0) = u 0 and some condition for the behavior as |x| → ∞, typically |u| → 0 which is made precise by considering solutions in certain functions spaces. The literature regarding solutions for these equations is quite large and we discuss only a small subset which is immediately relevant to this paper. A general open question for solutions is to discover conditions which guarantee a solution is "regular" (or smooth) for all time. For example, given an initial condition u 0 ∈ L 2 (R 3 ) Leray [12] proved there exists a solution (typically called a Leray-Hopf solution, see also ) u ∈ L ∞ ((0, ∞); L 2 (R 3 )) and ∇u ∈ L 2 ((0, ∞); L 2 (R 3 )). If u 0 is regular enough, the solution immediately enters the class of C ∞ smooth functions and remains there for some possibly small time (i.e. u ∈ C ∞ ((0, T * ) × R 3 ) but it remains an open question weather it retains this smoothness property for all time (i.e. u ∈ C ∞ ((0, ∞) × R 3 ).
Journal of Mathematical Analysis and Applications, 2009
We consider the evolution of a quantity advected by a compressible flow and subject to diffusion.... more We consider the evolution of a quantity advected by a compressible flow and subject to diffusion. When this quantity is scalar it can be, for instance, the temperature of the flow or the concentration of some pollutants. Because of the diffusion term, one expects the equations to have a regularizing effect. However, in their Euler form, the equations describe the evolution of the quantity multiplied by the density of the flow. The parabolic structure is thus degenerate near vacuum (when the density vanishes). In this paper we show that we can nevertheless derive uniform L p bounds that do not depend on the density (in particular the bounds do not degenerate near vacuum). Furthermore the result holds even when the density is only a measure.
Journal of Hyperbolic Differential Equations, 2009
We study a mathematical model for sprays which takes into account particle break-up due to drag f... more We study a mathematical model for sprays which takes into account particle break-up due to drag forces. In particular, we establish the existence of global weak solutions to a system of incompressible Navier-Stokes equations coupled with a Boltzmann-like kinetic equation. We assume the particles initially have bounded radii and bounded velocities relative to the gas, and we show that those bounds remain as the system evolves. One interesting feature of the model is the apparent accumulation of particles with arbitrarily small radii. As a result, there can be no nontrivial hydrodynamical equilibrium for this system.
Journal of Differential Equations, 2009
The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynami... more The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations is proved. The model consists of the conservation laws for the particle density and particle current density, including quantum corrections from the Bohm potential and viscous stabilizations arising from quantum Fokker-Planck interaction terms in the Wigner equation. The model equations are coupled self-consistently to the Poisson equation for the electric potential and are supplemented with periodic boundary and initial conditions. When a diffusion term linearly proportional to the velocity is introduced in the momentum equation, the positivity of the particle density is proved. This term, which introduces a strong regularizing effect, may be viewed as a classical conservative friction term due to particle interactions with the background temperature. Without this regularizing viscous term, only the nonnegativity of the density can be shown. The existence proof relies on the Faedo-Galerkin method together with a priori estimates from the energy functional.
Journal of Computational Physics, 2010
We present a generalized discontinuous Galerkin method for a multicomponent compressible barotrop... more We present a generalized discontinuous Galerkin method for a multicomponent compressible barotropic Navier-Stokes system of equations.