On the global large regular solutions of the 1D degenerate compressible Navier-Stokes equations

Abstract When the viscosity coefficients depend on the mass density <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ρ</m:mi> </m:math> \rho in the power <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>ρ</m:mi> </m:mrow> <m:mrow> <m:mi>δ</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>δ</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {\rho }^{\delta }\left(\delta \gt 0) , the existence of smooth solutions with vacuum of the compressible Navier-Stokes equations have received extensive attentions in recent years. In Cao et al. ( Global regular solutions for one-dimensional degenerate compressible Navier-Stokes equations with large data and far field vacuum , SIAM J. Math. Anal. 54 (2022), 4658–4694), when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>δ</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:math> 0\lt \delta \le 1 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>γ</m:mi> <m:mo>≥</m:mo> <m:mi>δ</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:math> \gamma \ge \delta +\frac{1}{2} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>γ</m:mi> </m:math> \gamma : the adiabatic exponent), the authors have established the global existence of a unique regular solution to the Cauchy problem of compressible Navier-Stokes equations with far field vacuum and general large data. In this article, via establishing some new accurate estimates of the effective velocity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>v</m:mi> </m:math> v and the velocity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> </m:math> u , we remove the crucial assumption <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>γ</m:mi> <m:mo>≥</m:mo> <m:mi>δ</m:mi> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:math> \gamma \ge \delta +\frac{1}{2} in Cao-Li-Zhu’s work, and only require that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>γ</m:mi> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:math> \gamma \gt 1 as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>δ</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:math> 0\lt \delta \le 1 . This improvement allows us to study more physical fluids including air. Moreover, the global existence obtained here does not depend on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>ρ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {\rho }_{0}\in {L}^{1} .