Singularity Formation for the Compressible Euler Equations

Abstract

It is well-known that singularity will develop in finite time for hyperbolic conservation laws from initial nonlinear compression no matter how small and smooth the data are. Classical results, including [P. Lax, J. Math. Phys., 5 (1964), pp. 611--614], [F. John, Comm. Pure Appl. Math., 27 (1974), pp. 377--405], [T. Liu, J. Differential Equations, 33 (1979), pp. 92--111], [T. Li, Y. Zhou, and D. Kong, Comm. Partial Differential Equations, 19 (1994), pp. 1263--1317], confirm that when initial data are small smooth perturbations near constant states, blowup in gradient of solutions occurs in finite time if initial data contain any compression in some truly nonlinear characteristic field, under some structural conditions. A natural question is, Will this picture keep true for large data problem of physical systems such as compressible Euler equations? One of the key issues is how to find an effective way to obtain sharp enough control on the density lower bound, which is known to decay to zero as time goes to infinity for a certain class of solutions. In this paper, we offer a simple way to characterize the decay of the density lower bound in time and therefore successfully classify the questions on singularity formation in compressible Euler equations. For isentropic flow, we offer a complete picture on the finite time singularity formation from smooth initial data away from vacuum, which is consistent with the small data theory. For adiabatic flow, we show a striking observation that initial weak compressions do not necessarily develop singularity in finite time. Furthermore, we follow [G. Chen, R. Young, and Q. Zhang, J. Hyperbolic Differ. Equ., 10 (2013), pp. 149--172] to introduce the critical strength of nonlinear compression and prove that if the compression is stronger than this critical value, then singularity develops in finite time, and otherwise there is a class of initial data admitting global smooth stationary solutions with maximum strength of compression equal to this critical value.