Stability of Planar Rarefaction Wave to Two-Dimensional Compressible Navier--Stokes Equations

It is well known that the rarefaction wave, one of the basic wave patterns of the hyperbolic conservation laws, is nonlinearly stable to the one-dimensional compressible Navier--Stokes equations (cf. [A. Matsumura and K. Nishihara, Japan J. Appl. Math., 3 (1986), pp. 1--13; Comm. Math. Phys., 144 (1992), pp. 325--335; T.-P. Liu and Z. P. Xin, Comm. Math. Phys., 118 (1988), pp. 451--465; K. Nishihara, T. Yang, and H. Zhao, SIAM J. Math. Anal., 35 (2004), pp. 1561--1597]). In the present paper we proved the time-asymptotical nonlinear stability of the planar rarefaction wave to the two-dimensional compressible and isentropic Navier--Stokes equations, which gives the first stability result of the planar rarefaction wave to the multidimensional system with physical viscosities.