Existence of global strong solution for the compressible Navier–Stokes equations with degenerate viscosity coefficients in 1D

Abstract

We consider Navier–Stokes equations for compressible viscous fluids in the one‐dimensional case. We prove the existence of global strong solution with large initial data for compressible Navier–Stokes equation with viscosity coefficients of the form ∂xfalse(ρα∂xufalse) with 12<α≤1 (it includes in particular the important physical case of the viscous shallow water system when α=1). The key ingredient of the proof relies to a new formulation of the compressible equations involving a new effective velocity v (see ) such that the density verifies a parabolic equation. We estimate v in Lt,x∞ norm which enables us to control the Lt,x∞ norm of 1ρ by using the maximum principle.

New effective pressure and existence of global strong solution for compressible Navier–Stokes equations with general viscosity coefficient in one dimension

“…This derivation relies on the hydrostatic approximation where the authors follow the role of viscosity and friction on the bottom. We are now going to rewrite the system (1.1) following the new formulation proposed in [11] (see also [9,8,7]), indeed setting:…”

“…we will generalize these techniques in the present paper to the case of general viscosity coefficients. In [10], the second author proved also the existence of global weak solution for general viscosity coefficients with initial density admitting shocks and with initial velocity belonging to the set of finite measures. In opposite to [14], the initial data satisfy the BD entropy but not the classical energy, it allows in particular to show some regularizing effects on the density inasmuch as the density becomes instantaneously continuous.…”

One-dimensional compressible heat-conducting gas with temperature-dependent viscosity

“…Remark 1.4. The existence of global strong solutions to the one-dimensional compressible Navier-Stokes equations for isentropic flows has been established in [27] with the viscosity µ given by (1.9)-(1.10) for ℓ 1 = 0, 0 ≤ ℓ 2 < 1 2 , and α = 0, and also in [11] for the shallow water system, where the viscosity µ satisfies (1.9) with h(v) = v −1 and α = 0. Note that our derivation of the uniform bounds on v(t, x) and θ(t, x) relies heavily on the assumption that the initial data is sufficiently smooth.…”

“…Note that our derivation of the uniform bounds on v(t, x) and θ(t, x) relies heavily on the assumption that the initial data is sufficiently smooth. It is an interesting and difficult problem to extend the results in [11] and [27] to the non-isentropic case with transport coefficients satisfying (1.9) for nonzero α. Now we outline the main ideas to deduce our main result Theorem 1.…”

Global Smooth Solutions for 1D Barotropic Navier–Stokes Equations with a Large Class of Degenerate Viscosities

“…The result of Mellet-Vasseur [15] was extended by Haspot [7] to the case of α ∈ (1/2, 1]. Recently, Constantin-Drivas-Nguyen-Pasqualotto [5, Theorem 1.6] extended it to the case of α ≥ 0 and γ ∈ [α, α + 1] with γ > 1, but they dealt with it on the periodic domain, and with an additional technical condition (see (1.6)).…”