Shock Formation in the Compressible Euler Equations and Related Systems

Abstract

We prove shock formation results for the compressible Euler equations and related systems of conservation laws in one space dimension, or three dimensions with spherical symmetry. We establish an L ∞ bound for C 1 solutions of the one-D Euler equations, and use this to improve recent shock formation results of the authors. We prove analogous shock formation results for one-D MHD with orthogonal magnetic field, and for compressible flow in a variable area duct, which has as a special case spherically symmetric three dimensional flow on the exterior of a ball.2010 Mathematical Subject Classification: 76N15, 35L65, 35L67.in the absence of regularizing dissipative effects such as viscosity and heat loss, is the blowup of gradients and subsequent formation of shock waves.

Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves

“…A far-from-exhaustive list of examples is: the groundbreaking work of Riemann [54] mentioned at the beginning, Lax's seminal finite-time breakdown results [43] for scalar conservation laws and his aforementioned application of the method of Riemann invariants to 2 × 2 genuinely nonlinear strictly hyperbolic systems [42], Jeffrey's work [26] on magnetoacoustics, JeffreyKorobeinikov's work [24] on nonlinear electromagnetism, Jeffrey-Teymur's work [25] on hyperelastic solids, John's extension [28] of Lax's work to systems in one spatial dimension with more than two unknowns (which required the development of new methods, in particularly identifying the important role played by simple waves, since the method of Riemann invariants is no longer applicable), Liu's further refinement [46] of John's work, John's work [30] on spherically symmetric solutions to the equations of elasticity, Klainerman-Majda's work [34] on nonlinear vibrating string equations, Bloom's work [5] on nonlinear electrodynamics, and Cheng-YoungZhang's work [10] on magnetohydrodynamics and related systems. Roughly, the blowup in all of these works is proved by finding a quantity y(t) that verifies a Riccatitype equationẏ(t) = a(t)y 2 (t) + Error, where a(t) is non-integrable in time near ∞ and Error is a small error term that does not interfere with the blowup.…”

“…9 Throughout we use Einstein's summation convention. 10 Note that ∂ t is not the same as the geometric coordinate partial derivative ∂ ∂t appearing in equation (2.22) and elsewhere throughout the article. is a parameter, fixed until Theorem 15.1 (our main theorem), and the data are nontrivial in the region {1 − U 0 ≤ x 1 ≤ 1} ∩ 0 := U 0 0 of thickness U 0 .…”

“…2.2), we have g αβ ( ) = m αβ +O( ), where m αβ = diag(−1, 1, 1) is the standard Minkowski metric and O( ) is an error term, smooth in and | | in magnitude when | | is small. Above and throughout, ∂ 0 , ∂ 1 , and ∂ 2 denote the corresponding rectangular coordinate partial derivatives, x 0 is alternate notation for the time coordinate t, and similarly 10 ∂ t := ∂ 0 . We make further mild assumptions on the nonlinearities ensuring that relative to rectangular coordinates, the nonlinear terms are effectively quadratic and fail to satisfy Klainerman's null condition [40]; see Subsect.…”

Optimal time-dependent lower bound on density for classical solutions of 1-D compressible Euler equations

“…The reason why we use ρ ε α and ρ ε β instead of α and β is to control the lower order terms in the Riccati equations produced by the varying entropy. The proof of Theorem 2.3 also relies on the uniform constant upper bound of density established in [8] by R. Young, Q. Zhang and the author for classical solutions when total variation of initial entropy is finite. One direct application of Theorem 2.3 is that one can use (2.13) to improve the life-span estimate established in [4] when 1 < γ < 3 which depends on the timedependent lower bound of density.…”

Optimal density lower bound on nonisentropic gas dynamics

“…In this paper, we will prove a lower bound on density in the optimal order O(1 + t) −1 for the non-isentropic solutions. The key new idea is to consider the following gradient variables, transformed from α and β in (2.1)-(2.2): Using this result, we can find constant upper bounds on α(x, t) and β(x, t) by the constant global upper bound on η given in [5], although max x∈R { α(x, t), β(x, t)} might not decay with respect to t. Then we can prove Theorem 1.1 using (2.4) and the initial constant lower bound on density (uniform upper bound on τ (x, 0)).…”