Global well-posedness of the 1d compressible Navier-Stokes system with rough data

In this paper, we study the global well-posedness problem for the 1d compressible Navier-Stokers system (cNSE) in gas dynamics with rough initial data. Frist, Liu- Yu (2022) established the global well-posedness theory for the 1d isentropic cNSE with initial velocity data in BV space. Then, it was extended to the 1d cNSE for the polytropic ideal gas with initial velocity and temperature data in BV space by Wang-Yu-Zhang (2022). We improve the global well-posedness result of Liu-Yu with initial velocity data in $W^{2\gamma ,1}$ space; and of Wang-Yu-Zhang with initial velocity data in $L^2\cap W^{2\gamma ,1}$ space and initial data of temperature in ${\dot{W}}^{-\frac{2}{3},\frac{6}{5}}\cap {\dot{W}}^{2\gamma -1,1}$ for any $\gamma >0$ $\mathit{\text{arbitrary}}\mathit{\text{small}}$. Our essential ideas are based on establishing various “end-point” smoothing estimates for the 1d parabolic equation.