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Steve Shkoller - One of the best experts on this subject based on the ideXlab platform.
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well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum
Archive for Rational Mechanics and Analysis, 2012Co-Authors: Daniel Coutand, Steve ShkollerAbstract:We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = Cγργ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial Viscosity Term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial Viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial Viscosity tends to zero. Our regular solutions can be viewed as degenerate Viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.
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well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum
Communications on Pure and Applied Mathematics, 2011Co-Authors: Daniel Coutand, Steve ShkollerAbstract:We prove well-posedness for the 3-D compressible Euler equations with moving physical vacuum boundary, with an equation of state given by the so-called gamma gas-law for gamma > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss--Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial Viscosity Term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a new higher-order Hardy-type inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial Viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial Viscosity tends to zero. Our regular solutions can be viewed as degenerate Viscosity solutions. Out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.
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The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations
Archive for Rational Mechanics and Analysis, 2006Co-Authors: Daniel Coutand, Steve ShkollerAbstract:The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations. Unlike our approach in [5] for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial Viscosity Term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial Viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in strong norms of our sequence of regularized problems.
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on the interaction between quasilinear elastodynamics and the navier stokes equations
arXiv: Analysis of PDEs, 2005Co-Authors: Daniel Coutand, Steve ShkollerAbstract:The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations along a moving interface. Unlike our approach for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial Viscosity Term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial Viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in srong norms of our sequence of regularized problems.
Daniel Coutand - One of the best experts on this subject based on the ideXlab platform.
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well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum
Archive for Rational Mechanics and Analysis, 2012Co-Authors: Daniel Coutand, Steve ShkollerAbstract:We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = Cγργ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial Viscosity Term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial Viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial Viscosity tends to zero. Our regular solutions can be viewed as degenerate Viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.
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well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum
Communications on Pure and Applied Mathematics, 2011Co-Authors: Daniel Coutand, Steve ShkollerAbstract:We prove well-posedness for the 3-D compressible Euler equations with moving physical vacuum boundary, with an equation of state given by the so-called gamma gas-law for gamma > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss--Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial Viscosity Term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a new higher-order Hardy-type inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial Viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial Viscosity tends to zero. Our regular solutions can be viewed as degenerate Viscosity solutions. Out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.
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The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations
Archive for Rational Mechanics and Analysis, 2006Co-Authors: Daniel Coutand, Steve ShkollerAbstract:The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations. Unlike our approach in [5] for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial Viscosity Term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial Viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in strong norms of our sequence of regularized problems.
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on the interaction between quasilinear elastodynamics and the navier stokes equations
arXiv: Analysis of PDEs, 2005Co-Authors: Daniel Coutand, Steve ShkollerAbstract:The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations along a moving interface. Unlike our approach for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial Viscosity Term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial Viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in srong norms of our sequence of regularized problems.
Guiqiang Chen - One of the best experts on this subject based on the ideXlab platform.
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vanishing Viscosity limit of the navier stokes equations to the euler equations for compressible fluid flow
Communications on Pure and Applied Mathematics, 2010Co-Authors: Guiqiang Chen, Mikhail PerepelitsaAbstract:We establish the vanishing Viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical Viscosity Term so that the uniform sup-norm of solutions with respect to the physical Viscosity coefficient may not be directly controllable. Furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the Viscosity coefficient for solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported C2 test functions, are confined in a compact set in H−1, which leads to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations with finite-energy initial data, relative to the different end-states at infinity. © 2010 Wiley Periodicals, Inc.
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vanishing Viscosity limit of the navier stokes equations to the euler equations for compressible fluid flow
arXiv: Analysis of PDEs, 2009Co-Authors: Guiqiang Chen, Mikhail PerepelitsaAbstract:We establish the vanishing Viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical Viscosity Term so that the uniform sup-norm of solutions with respect to the physical Viscosity coefficient may not be directly controllable and, furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the Viscosity coefficient for the solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported $C^2$ test functions, are confined in a compact set in $H^{-1}$, which lead to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of the measure-valued solution to a Delta mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations.
Mikhail Perepelitsa - One of the best experts on this subject based on the ideXlab platform.
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vanishing Viscosity limit of the navier stokes equations to the euler equations for compressible fluid flow
Communications on Pure and Applied Mathematics, 2010Co-Authors: Guiqiang Chen, Mikhail PerepelitsaAbstract:We establish the vanishing Viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical Viscosity Term so that the uniform sup-norm of solutions with respect to the physical Viscosity coefficient may not be directly controllable. Furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the Viscosity coefficient for solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported C2 test functions, are confined in a compact set in H−1, which leads to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations with finite-energy initial data, relative to the different end-states at infinity. © 2010 Wiley Periodicals, Inc.
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vanishing Viscosity limit of the navier stokes equations to the euler equations for compressible fluid flow
arXiv: Analysis of PDEs, 2009Co-Authors: Guiqiang Chen, Mikhail PerepelitsaAbstract:We establish the vanishing Viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical Viscosity Term so that the uniform sup-norm of solutions with respect to the physical Viscosity coefficient may not be directly controllable and, furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the Viscosity coefficient for the solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported $C^2$ test functions, are confined in a compact set in $H^{-1}$, which lead to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of the measure-valued solution to a Delta mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations.
Jaime Peraire - One of the best experts on this subject based on the ideXlab platform.
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Sub-Cell Shock Capturing for Discontinuous Galerkin Methods
44th AIAA Aerospace Sciences Meeting and Exhibit, 2006Co-Authors: Per O. Å. Persson, Jaime PeraireAbstract:A shock capturing strategy for higher order Discontinuous Galerkin approximations of scalar conservation laws is presented. We show how the original explicit artificial Viscosity methods proposed over fifty years ago for finite volume methods, can be used very effectively in the context of high order approximations. Rather than relying on the dissipation inherent in Discontinuous Galerkin approximations, we add an artificial Viscosity Term which is aimed at eliminating the high frequencies in the solution, thus eliminating Gibbs-type oscillations. We note that the amount of Viscosity required for stability is deTermined by the resolution of the approximating space and therefore decreases with the order of the approximating polynomial. Unlike classical finite volume artificial Viscosity methods, where the shock is spread over several computational cells, we show that the proposed approach is capable of capturing the shock as a sharp, but smooth profile, which is typically contained within one element. The method is complemented with a shock detection algorithm which is based on the rate of decay of the expansion coefficients of the solution when this is expressed in a hierarchical orthonormal basis. For the Euler equations, we consider and discuss the performance of several forms of the artificial Viscosity Term.
