On Classical Solutions for Viscous Polytropic Fluids with Degenerate Viscosities and Vacuum

Abstract

In this paper, we consider the three-dimensional isentropic Navier-Stokes equations for compressible fluids allowing initial vacuum when viscosities depend on density in a superlinear power law. We introduce the notion of regular solutions and prove the local-in-time well-posedness of solutions with arbitrarily large initial data and vacuum in this class, which is a long-standing open problem due to the very high degeneracy caused by vacuum. Moreover, for certain classes of initial data with local vacuum, we show that the regular solution that we obtained will break down in finite time, no matter how small and smooth the initial data are.

Vanishing Viscosity Limit of the Navier–Stokes Equations to the Euler Equations for Compressible Fluid Flow with Vacuum

“…Actually, our desired a priori estimates mainly come from the "quasi-symmetric hyperbolic-"degenerate elliptic" coupled structure (1.19) 2 , and the details could be seen in Section 3. Moreover, we point out that the regular solution obtained in the above theorem will break down in finite time, if the initial data contain "isolated mass group" or "hyperbolic singularity set", which could be rigorously proved via the same arguments used in [30]. Letting ǫ → 0, the solution obtained in Theorem 1.2 will strongly converge to that of the compressible Euler equations (1.9) in C([0, T ]; H s ′ ) for any s ′ ∈ [1,3).…”

“…In particular, these systems become highly degenerate, which results in that the velocity can not even be defined in the vacuum domain and hence it is difficult to get uniform estimates for the velocity near the vacuum. Recently in Li-Pan-Zhu [29,30], via carefully analyzing the mathematical structure of these systems, they reasonably gave the time evolution mechanism of the fluid velocity in the vacuum domain. Taking δ = 1 for example, via considering the following parabolic equations with a special source term:…”

“…Based on the well-posedness theory mentioned above, naturally there is an important question that: can we regard the regular solution of inviscid flow [34,39] as those of viscous flow [10,14,23,29,30] with vanishing real physical viscosities?…”

Vanishing viscosity limit of the rotating shallow water equations with far field vacuum

“…When viscosity coefficients are height-dependent, Li-Pan-Zhu [12,14,20,21] proposed a new quantity ∇ρ/ρ, which should belong to space L 6 ∩ D 1 ∩ D 2 , to obtain the local existence of classical solutions with far vacuum. Some other results on degenerate viscosities and initial vacuum can be seen in [13,15]. Recently, via introducing a symmetric structure for the quantity ∇ρ/ρ and a "quasi-symmetric hyperbolic"-"elliptic" coupled structure for (ρ, u), Ding-Zhu [6] showed the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for the compressible fluid with far field vacuum.…”

Finite-time blow-up phenomena of Vlasov/Navier–Stokes equations and related systems

“…For non-compactly supported initial data, blow up estimates are provided in [27,34] under certain decay assumptions on solutions at far fields. More recently, the finite time singularity formation of regular solutions for viscous compressible fluids without heat conduction is stuided in [28,39] for the initial data with isolated mass group.…”