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The large time stability of sound waves. (English) Zbl 0858.76075
We demonstrate the existence of solutions to the full \(3 \times 3\) system of compressible Euler equations in one space dimension, up to an arbitrary time \(T>0\), in the case when the initial data have arbitrarily large total variation, and sufficiently small supnorm. The result applies to periodic solutions of the Euler equations, a nonlinear model for sound wave propagation in gas dynamics. Our analysis establishes a growth rate for the total variation that depends on a new length scale \(d\) that we identify in the problem. In the limit \(d \to \infty\), we recover Glimm’s theorem, and we observe that there exist linearly degenerate systems within the class considered for which the growth rate we obtain is sharp.

MSC:
76Q05 Hydro- and aero-acoustics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics
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