The Experts below are selected from a list of 315 Experts worldwide ranked by ideXlab platform

Guiqiang Chen - One of the best experts on this subject based on the ideXlab platform.

  • stability of steady multi wave configurations for the full euler equations of Compressible Fluid Flow
    Acta Mathematica Scientia, 2018
    Co-Authors: Guiqiang Chen, Matthew Rigby
    Abstract:

    Abstract We are concerned with the stability of steady multi-wave configurations for the full Euler equations of Compressible Fluid Flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the Flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic Flow are stable in structure globally, even under the BV perturbation of the incoming Flow in the Flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the Flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.

  • vanishing viscosity limit of the navier stokes equations to the euler equations for Compressible Fluid Flow
    Communications on Pure and Applied Mathematics, 2010
    Co-Authors: Guiqiang Chen, Mikhail Perepelitsa
    Abstract:

    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.

  • Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for Compressible Fluid Flow
    Communications on Pure and Applied Mathematics, 2010
    Co-Authors: Guiqiang Chen, Mikhail Perepelitsa
    Abstract:

    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.

  • vanishing viscosity limit of the navier stokes equations to the euler equations for Compressible Fluid Flow
    arXiv: Analysis of PDEs, 2009
    Co-Authors: Guiqiang Chen, Mikhail Perepelitsa
    Abstract:

    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.

Yu Zhang - One of the best experts on this subject based on the ideXlab platform.

  • Concentration and cavitation in the vanishing pressure limit of solutions to the generalized Chaplygin Euler equations of Compressible Fluid Flow
    European Journal of Mechanics B-fluids, 2019
    Co-Authors: Yu Zhang, Yicheng Pang, Jinhuan Wang
    Abstract:

    Abstract The phenomena of concentration and cavitation and the formation of delta shock waves and vacuum states in vanishing pressure limits of solutions to the generalized Chaplygin Euler equations of Compressible Fluid Flow are analyzed. It is proved that, as the pressure vanishes, any two-shock-wave Riemann solution of the generalized Chaplygin Euler equations of Compressible Fluid Flow tends to a delta-shock solution to the transport equations, and the intermediate density between them tends to a weighted δ -measure that forms a delta shock wave; any two-rarefaction-wave solution is shown to tend to two contact discontinuities connecting the constant states and vacuum states, which form a vacuum solution of the transport equations. Moreover, some numerical simulations completely coinciding with the theoretical analysis are presented.

  • delta shock wave to the Compressible Fluid Flow with the generalized chaplygin gas
    International Journal of Non-linear Mechanics, 2018
    Co-Authors: Yicheng Pang, Yu Zhang
    Abstract:

    Abstract We concern with the Riemann problem the Compressible Fluid Flow with the generalized Chaplygin gas. With the analysis on the phase plane, we rigorously confirm the occurrence of delta shock wave with Dirac delta function in density. Then the formation mechanism, generalized Rankine–Hugoniot relation and entropy condition for the delta shock wave are clarified. Based on these preparations, five kinds of exact solutions are obtained. Finally, the corresponding numerical results are also presented to illustrate our analysis.

Stephen C Anco - One of the best experts on this subject based on the ideXlab platform.

  • conserved integrals for inviscid Compressible Fluid Flow in riemannian manifolds
    Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2015
    Co-Authors: Stephen C Anco, Nazim Tufail
    Abstract:

    An explicit determination of all local conservation laws of kinematic type on moving domains and moving surfaces is presented for the Euler equations of inviscid Compressible Fluid Flow in curved R...

  • conservation laws of inviscid non isentropic Compressible Fluid Flow in n 1 spatial dimensions
    Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2010
    Co-Authors: Stephen C Anco
    Abstract:

    Recent work giving a classification of kinematic and vorticity conservation laws of Compressible Fluid Flow with barotropic equations of state (where pressure is a function only of the Fluid density) in n >1 spatial dimensions is extended to general non-isentropic equations of state in which the pressure is also a function of the dynamical entropy (per unit mass) of the Fluid. Two main results are obtained. First, we find that, apart from the familiar conserved integrals for mass, momentum, energy, angular momentum, Galilean momentum and volumetric entropy, additional kinematic conserved integrals arise only for non-isentropic equations of state given by a generalized form of the well-known polytropic equation of state with dimension-dependent exponent γ =1+2/ n , such that the proportionality coefficient is an arbitrary function of the entropy (per unit mass). Second, we show that the only vorticity conserved integrals consist of a circulatory entropy (which vanishes precisely when the Fluid Flow is irrotational) in all even dimensions. In particular, the vorticity integrals for helicity in odd dimensions and enstrophy in even dimensions are found to be no longer conserved for any non-isentropic equation of state.

Mikhail Perepelitsa - One of the best experts on this subject based on the ideXlab platform.

  • vanishing viscosity limit of the navier stokes equations to the euler equations for Compressible Fluid Flow
    Communications on Pure and Applied Mathematics, 2010
    Co-Authors: Guiqiang Chen, Mikhail Perepelitsa
    Abstract:

    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.

  • Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for Compressible Fluid Flow
    Communications on Pure and Applied Mathematics, 2010
    Co-Authors: Guiqiang Chen, Mikhail Perepelitsa
    Abstract:

    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.

  • vanishing viscosity limit of the navier stokes equations to the euler equations for Compressible Fluid Flow
    arXiv: Analysis of PDEs, 2009
    Co-Authors: Guiqiang Chen, Mikhail Perepelitsa
    Abstract:

    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.

Zhiqiang Shao - One of the best experts on this subject based on the ideXlab platform.

  • concentration of mass in the pressureless limit of the euler equations of one dimensional Compressible Fluid Flow
    Nonlinear Analysis-real World Applications, 2020
    Co-Authors: Shouqiong Sheng, Zhiqiang Shao
    Abstract:

    Abstract In this paper, we study the limiting behavior of Riemann solutions to the Euler equations of one-dimensional Compressible Fluid Flow as γ tends to one. We show that the limit solution forms the delta wave to the pressureless Euler system of one-dimensional Compressible Fluid Flow in the distribution sense. Some numerical results exhibiting the phenomena of concentration are also presented.

  • The pressureless limits of Riemann solutions to the Euler equations of one-dimensional Compressible Fluid Flow with a source term
    arXiv: Analysis of PDEs, 2019
    Co-Authors: Shouqiong Sheng, Zhiqiang Shao
    Abstract:

    In this paper, we study the limits of Riemann solutions to the inhomogeneous Euler equations of one-dimensional Compressible Fluid Flow as the adiabatic exponent $\gamma$ tends to one. Different from the homogeneous equations, the Riemann solutions of the inhomogeneous system are non self-similar. It is rigorously shown that, as $\gamma$ tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a source term, and the intermediate density between the two shocks tends to a weighted $\delta$-mesaure which forms the delta shock; while any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution of the pressureless Euler system with a source term, whose intermediate state between the two contact discontinuities is a vacuum state. Moreover, we also give some numerical results to confirm the theoretical analysis.