Commun. Math. Anal. Appl., 2 (2023), pp. 21-69.
Published online: 2023-03
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The vanishing viscosity limit of the three-dimensional (3D) compressible and isentropic Navier-Stokes equations is proved in the case that the corresponding 3D inviscid Euler equations admit a planar rarefaction wave solution connected with vacuum states. Moreover, a uniform convergence rate with respect to the viscosity coefficients is obtained. Compared with previous results on the zero dissipation limit to planar rarefaction wave away from vacuum states [27, 28], the new ingredients and main difficulties come from the degeneracy of vacuum states in the planar rarefaction wave in the multi-dimensional setting. Suitable cut-off techniques and some delicate estimates are needed near the vacuum states. The inviscid decay rate around the planar rarefaction wave with vacuum is determined by the cut-off parameter and the nonlinear advection flux terms of 3D compressible Navier-Stokes equations.
}, issn = {2790-1939}, doi = {https://doi.org/ 10.4208/cmaa.2022-0020}, url = {http://global-sci.org/intro/article_detail/cmaa/21453.html} }The vanishing viscosity limit of the three-dimensional (3D) compressible and isentropic Navier-Stokes equations is proved in the case that the corresponding 3D inviscid Euler equations admit a planar rarefaction wave solution connected with vacuum states. Moreover, a uniform convergence rate with respect to the viscosity coefficients is obtained. Compared with previous results on the zero dissipation limit to planar rarefaction wave away from vacuum states [27, 28], the new ingredients and main difficulties come from the degeneracy of vacuum states in the planar rarefaction wave in the multi-dimensional setting. Suitable cut-off techniques and some delicate estimates are needed near the vacuum states. The inviscid decay rate around the planar rarefaction wave with vacuum is determined by the cut-off parameter and the nonlinear advection flux terms of 3D compressible Navier-Stokes equations.