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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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##### References:
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