2019/01/10 by Boris Haspot, Haspot, Boris
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #Cosmology and Gravitation Theories #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1901.03143
openalex publication_date 2019/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we investigate the question of the existence of global weak\nsolution for the compressible Navier Stokes equations provided that the initial\nmomentum \ρ0 u0 belongs to \bmo-1(\ℝN) with N= 2,3\nand is radially symmetric. More precisely we deal with the so called viscous\nshallow water system when the viscosity coefficients verify\n\μ(\ρ)=\μ\ρ, \λ(\ρ)=0 with \μ>0. We prove then a equivalent\nof the so called Koch-Tataru theorem for the compressible Navier-Stokes\nequations. In addition we assume that the initial density \ρ0 is only\nbounded in L^\∞(\ℝN), it allows us in particular to consider\ninitial density admitting shocks. Furthermore we show that if the coupling\nbetween the density and the velocity is sufficiently strong, then the initial\ndensity which admits initially shocks is instantaneously regularizing inasmuch\nas the density becomes Lipschitz. This coupling is expressed via the regularity\nof the so called effective velocity v=u+2\μ\∇\ln\ρ. In our case v0\nbelongs to L2(\ℝN)\∩ L^\∞(\ℝN), it is important to\npoint out that this choice on the initial data implies that we work in a\nsetting of infinite energy on the initial data (\ρ0,u0), it extends in\nparticular the results of citeV. In a similar way, we consider also the case\nof the dimension N=1 where the momentum \ρ0 u0 belongs to\nbmo-1(\ℝ) without any geometric restriction. To finish we prove\nthe global existence of strong solution for large initial data provided that\nthe initial data are radially symmetric and sufficiently regular in dimension\nN=2,3 for \γ law pressure.\n