Entropy‐Bounded Solutions to the One‐Dimensional Heat Conductive Compressible Navier‐Stokes Equations with Far Field Vacuum

Abstract

In the presence of vacuum, the physical entropy for polytropic gases behave singularly, and it is thus a challenge to study its dynamics. It is shown in this paper that the boundedness of the entropy can be propagated up to any finite time provided that the initial vacuum presents only at far fields with sufficiently slow decay of the initial density. More precisely, for the Cauchy problem of the one-dimensional heat conductive compressible Navier-Stokes equations, the global well-posedness of strong solutions and uniform boundedness of the corresponding entropy are established, as long as the initial density vanishes only at far fields with a rate no more than O. 1x 2 /. The main tools of proving the uniform boundedness of the entropy are some singularly weighted energy estimates carefully designed for the heat conductive compressible Navier-Stokes equations and an elaborate De Giorgi type iteration technique for some classes of degenerate parabolic equations. The De Giorgi-type iterations are carried out to different equations in establishing the lower and upper bounds of the entropy.

Local well-posedness of isentropic compressible Navier-Stokes equations with vacuum

“…In the absence of vacuum, that is the density has positive lower bound, the system is locally well-posed for large initial data, see, e.g., [23,41,46,48,49,51]; however, the global well-posedness is still unknown. It has been known that system in one dimension is globally well-posed for large initial data, see, e.g., [2, 26-29, 57, 58] and the references therein, see [35] for the large time behavior of the solutions, and also [34,37,38] for the global well-posedness for the case that with nonnegative density. For the multi-dimensional case, the global well-posedness holds for small initial data, see, e.g., [6, 10-13, 19, 30, 42-45, 47, 50].…”

Propagation of uniform boundedness of entropy and inhomogeneous regularities for viscous and heat conductive gases with far field vacuum in three dimensions

“…Further more, the blowup results of Xin [52] and Xin-Yan [53] imply that the global solutions established in [19,28,51] must have unbounded entropy, if initially there is an isolated mass group surrounded by the vacuum region. However, it is somewhat surprisingly that if the initial density vanishes only at far fields with a rate no more than O( 1 |x| 2 ), then, as for the non-vacuum case, the solutions can be established in both the homogeneous and inhomogeneous spaces, and the entropy can be uniformly bounded up to any finite time, at least in the one-dimensional case, see the recent works by the authors [34,35]. Mathematically, since system (1.1)-(1.3) is already closed, one can establish the corresponding theory for it, regardless of the entropy.…”

“…Technically, due to the lack of the expression of the entropy in the vacuum region and the high singularity and degeneracy of the entropy equation near the vacuum region, in spite of its importance, the mathematical analysis of the entropy for the viscous compressible fluids in the presence of vacuum was rarely carried out before. Recently, we have initiated studies on these issues in the one dimensional case in [34,35]. It was proved in [34,35] that the one-dimensional compressible Navier-Stokes equations, with or without heat conducting, can propagate the uniform boundedness of the entropy locally or globally in time, as long as the initial density vanishes only at far fields with a rate no more than O( 1x 2 ).…”

“…Recently, we have initiated studies on these issues in the one dimensional case in [34,35]. It was proved in [34,35] that the one-dimensional compressible Navier-Stokes equations, with or without heat conducting, can propagate the uniform boundedness of the entropy locally or globally in time, as long as the initial density vanishes only at far fields with a rate no more than O( 1x 2 ). However, the problems in the multi-dimensional case have not been studied.…”

Global strong solutions to the Cauchy problem of the planar non-resistive magnetohydrodynamic equations with large initial data

“…Inspired by Li-Xin [25,26], in this paper, we will study the global existence and uniqueness of strong solution to the Cauchy problem for the planar MHD equations (1.3). The initial data is assumed to be large and may contain vacuum.…”