Department of Mathematics, University of California, Santa Barbara, California .... This is a consequence of local energy estimates; see the proposition in [10].
Communications in Commun. Math. Phys. 101, 475-485 (1985)
MathθΠΓiatlCal
Physics
© Springer-Verlag 1985
Formation of Singularities in Three-Dimensional Compressible Fluids Thomas C. Sideris Department of Mathematics, University of California, Santa Barbara, California 93106, USA
Abstract. Presented are several results on the formation of singularities in solutions to the three-dimensional Euler equations for a polytropic, ideal fluid under various assumptions on the initial data. In particular, it is shown that a localized fluid which is initially compressed and outgoing, on average, will develop singularities regardless of the size of the initial disturbance. This article presents a number of results on formation of singularities in solutions to the three-dimensional compressible Euler equations for a polytropic, ideal fluid. The results, described precisely in the following section, show that C 1 solutions to the equations do not exist for all time, under various restrictions on the initial data. Theorems one and two, below, deal with the case of "large" data which essentially means that the initial flow velocity must be supersonic in some region relative to the sound speed at infinity. (The initial data is constant outside a bounded set.) Singularity formation is detected as the disturbance overtakes the wave front (presumably as a shock wave) forcing the front to propagate with supersonic speed. The method, which is a refinement of [10], applies equally well in one and two space dimensions. Our main result, Theorem 3, shows that a fluid will develop singularities if, on average, it is slightly compressed and out-going near the wave front. The proof borrows some important technical points from an earlier result on nonlinear wave equations in three space dimensions [9]. Although the extensive one-dimensional theory [2,5,7,8] strongly suggests such results, the customary method of characteristics has not proved tractable in higher dimensions. (John [3] has recently used characteristics to prove blow-up for a threedimensional scalar wave equation with spherical symmetry.) Our approach involves the use of averaged quantities, thereby avoiding for the most part, local analysis of the solution. For classical solutions, these averages satisfy certain differential inequalities, solutions of which have finite life span. We do not address the problem of singularity formation in three-dimensional incompressible fluids. It is possible to approximate a compressible flow with an Research supported in part by the National Science Foundation
476
T. Sideris
incompressible one by letting the sound speed become large [6]. However, our upper bound for the life span of a C 1 compressible flow is not uniform in this parameter, and so this approximation cannot be used to establish singularity formation in the incompressible case. 1. Statement of Results The motion of a polytropic, ideal gas is described by the compressible Euler equations [1]: pt + V-pu = 0,
(1.1a)
p(ut + M VM) + Vp = 0,
(1.1b)
S ί + u VS = 0,
(1.1c)
(A>0,y>l).
(l.ld)
y
s
p = Ap e
Here, p, u, S, p represent the density, velocity, specific entropy, and pressure of the gas, respectively. The state Eq. (l.ld) is that of a polytropic gas, y being the adiabatic index. Initial data is assigned which is constant outside the bounded set {|x| ^ R}: p(x,0) = p°(x)>0;
p°(x) = p,
u(x,0) = u°(x);
u°(x) = ύ( = 0l
S(x,0) = S°(x);
S°(x) = S,
\x\ZR, |x|^R,
|x|^jR.
(1.2a) (1.2b) (1.2c)
By making the change of variables u-±u — ΰ and x->x + tΰ, we may assume without further loss of generality that ΰ = 0, as indicated in (1.2b). The Eqs. (1.1 a-d) can be written as a positive, symmetric hyperbolic system, and as such, they possess a unique, local C 1 solution (p,w, 0 , provided the initial data (1.2 a-c) are sufficiently regular [4]. The maximum speed of propagation of the front of a smooth disturbance is governed by the sound speed 1 s p
e y'\
(1.3)
since ΰ = 0. More precisely, letting
we have the following. Proposition. // (p, u, S) is a C 1 solution of (1.1 a-d), {1.2 a-c) on D(t), for 0 ^ t Ro + at) P(r, ή = P°(r, t) + — ]
j
ΠP(y, τ)dydτ
2 0, we have that J2 ^ C(σt + R)-2 J Oτ\ (tOσt + Ko
τ)(y - στ - R0)2G(y,
τ)dydτ,
for t^Rx. Returning to (3.3b), and noting that G(y,τ) vanishes for y > στ + JR, we can integrate by parts and then use (3.7) to obtain J2 ^ C(σt + RΓ2 f ΰτ]R (t - τ)G(y, τ)dydτ 0 στ + Ro 0-2 t
^C(σt + RΓ2—$(t-τ) P 0
στ + R
J
J ω(p-p)2dxdydτ.
(3.8a)
J ,
Denoting this last integral by J3 and using Schwartz's inequality, we find 2
j
στ + Ro
y (R — Ro)/2. In [9], we showed that any C function F(s) which satisfies the 2 above inequalities has a finite life span bounded above by C exp [Cp /B%] (cf. 42 a-d and the conclusion of the proof of Theorem 3 in [9]). 1 Hence, the life span of the C solution is bounded above by 2
2
-εxp(Cσ /B ). σ
For the general case, y > 1, the adjustment occurs in (3.7) where now p-pσ2(p - p)^ Aes~lpy-p?- yp'~\p - p)] = Aes~Ψ(p,p). (3.10) By convexity of py, Ψ(p, p) is positive for p Φ p. Using Taylor's theorem, we have the lower bound where Φy() is nonnegative, convex and a2, a\
— p
