×

Stability of planar rarefaction waves under general viscosity perturbation of the isentropic Euler system. (English) Zbl 1494.35132

The authors consider the vanishing viscosity limit for a model of a general non-Newtonian compressible fluid in \(\mathbb{R}^d\), \(d = 2, 3\). It is supposed that the initial data approach a profile determined by the Riemann data generating a planar rarefaction wave for the isentropic Euler system. The 1-D Euler system has the form: \[ \partial_t\tilde{\rho }+ \partial_{x_1}(\tilde{\rho }\tilde{u})=0, \] \[ \partial_t(\tilde{\rho }\tilde{u })+ \partial_{x_1}(\tilde{\rho }\tilde{u}^2) +\partial_{x_1}p(\tilde{\rho })=0,\quad x_1\in\mathbb{R} \] with the Riemann initial data
\([\tilde{\rho }(0,x_1),\tilde{u}(0,x_1)]= [\rho_0,\mathbf{u}_0]= [\tilde{\rho }_L,\tilde{u}_L]\) if \(x_1<0\) or \(=[\tilde{\rho }_R,\tilde{u}_R]\) if \(x_1\geq 0\), where \(\tilde{\rho }_L, \tilde{\rho }_R\), and \(\tilde{u}_L,\tilde{u}_R\in\mathbb{R}\) are constants. This Riemann problem admits a self-similar solution \(\tilde{\rho }(x_1/t)\), \(\tilde{u}(x_1/t)\). Moreover, the solution is Lipschitz for any \(t > 0\) provided some inequality hold given in a paper by E. Chiodaroli and O. Kreml [Arch. Ration. Mech. Anal. 214, No. 3, 1019–1049 (2014; Zbl 1304.35516)]. The rarefaction waves are actually the shock free solutions to the 1-D Riemann problem. Note that the rarefaction waves are stable objects. It is known that the solutions of the isentropic Navier-Stokes system converge to their Euler counterparts in the 1-D. There is a paper by L.-A. Li et al. [Commun. Math. Phys. 376, No. 1, 353–384 (2020; Zbl 1439.35372)], where the authors construct suitable smooth perturbations of the initial data, with the associated strong solutions converging to the rarefaction waves. The main result here concerns general not necessarily smooth and small perturbation of the initial data and rather the general class of viscous stress tensors with a nonlinear dependence on the velocity gradient. The associated sequence of dissipative solutions approaches the corresponding rarefaction wave strongly in the energy norm in the vanishing viscosity limit. The result covers the particular case of a linearly viscous fluid governed by the Navier-Stokes system.

MSC:

35Q31 Euler equations
76A05 Non-Newtonian fluids
35Q30 Navier-Stokes equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
76N30 Waves in compressible fluids
35C06 Self-similar solutions to PDEs
35D35 Strong solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abbatiello, A.; Feireisl, E.; Novotný, A., Generalized solutions to models of compressible viscous fluids, Discrete Contin. Dyn. Syst., Ser. A, 41, 1, 1-28 (2021) · Zbl 1460.35274
[2] Dafermos, C. M., The second law of thermodynamics and stability, Arch. Ration. Mech. Anal., 70, 167-179 (1979) · Zbl 0448.73004
[3] Chen, G.-Q.; Chen, J., Stability of rarefaction waves and vacuum states for the multidimensional Euler equations, J. Hyperbolic Differ. Equ., 4, 1, 105-122 (2007) · Zbl 1168.35312
[4] Chen, G.-Q.; Perepelitsa, M., Vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for compressible fluid flow, Commun. Pure Appl. Math., 63, 11, 1469-1504 (2010) · Zbl 1205.35188
[5] Chiodaroli, E.; De Lellis, C.; Kreml, O., Global ill-posedness of the isentropic system of gas dynamics, Commun. Pure Appl. Math., 68, 7, 1157-1190 (2015) · Zbl 1323.35137
[6] Chiodaroli, E.; Kreml, O., On the energy dissipation rate of solutions to the compressible isentropic Euler system, Arch. Ration. Mech. Anal., 214, 3, 1019-1049 (2014) · Zbl 1304.35516
[7] Chiodaroli, E.; Kreml, O.; Mácha, V.; Schwarzacher, S., Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data (2019), Archive Preprint Series
[8] Feireisl, E.; Ghoshal, S. S.; Jana, A., On uniqueness of dissipative solutions to the isentropic Euler system, Commun. Partial Differ. Equ., 44, 12, 1285-1298 (2019) · Zbl 1428.35325
[9] Feireisl, E.; Kreml, O., Uniqueness of rarefaction waves in multidimensional compressible Euler system, J. Hyperbolic Differ. Equ., 12, 3, 489-499 (2015) · Zbl 1327.35230
[10] Feireisl, E.; Kreml, O.; Vasseur, A., Stability of the isentropic Riemann solutions of the full multidimensional Euler system, SIAM J. Math. Anal., 47, 3, 2416-2425 (2015) · Zbl 1325.35148
[11] Feireisl, E.; Kwon, Y.-S.; Novotný, A., On the long-time behavior of dissipative solutions to models of non-Newtonian compressible fluids (2020), Archive Preprint Series
[12] Goodman, J.; Xin, Z. P., Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal., 121, 3, 235-265 (1992) · Zbl 0792.35115
[13] Hoff, D.; Liu, T.-P., The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data, Indiana Univ. Math. J., 38, 4, 861-915 (1989) · Zbl 0674.76047
[14] Kwon, Y. S.; Novotný, A., Dissipative solutions to compressible Navier-Stokes equations with general inflow-outflow data: existence, stability and weak-strong uniqueness, J. Math. Fluid Mech. (2021), (in press) · Zbl 1497.76068
[15] Li, L.-A.; Wang, D.; Wang, Y., Vanishing viscosity limit to the planar rarefaction wave for the two-dimensional compressible Navier-Stokes equations, Commun. Math. Phys., 376, 353-384 (2020) · Zbl 1439.35372
[16] Lions, P.-L., Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models (1998), Oxford Science Publication: Oxford Science Publication Oxford · Zbl 0908.76004
[17] Markfelder, S.; Klingenberg, C., The Riemann problem for the multidimensional isentropic system of gas dynamics is ill-posed if it contains a shock, Arch. Ration. Mech. Anal., 227, 3, 967-994 (2018) · Zbl 1390.35249
[18] Matsumura, A.; Nishihara, K., Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Jpn. J. Appl. Math., 3, 1, 1-13 (1986) · Zbl 0612.76086
[19] Perepelitsa, M., Asymptotics toward rarefaction waves and vacuum for 1-D compressible Navier-Stokes equations, SIAM J. Math. Anal., 42, 3, 1404-1412 (2010) · Zbl 1211.35189
[20] Xin, Z. P., Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Commun. Pure Appl. Math., 46, 5, 621-665 (1993) · Zbl 0804.35108
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.