| Abstract: |
| It is well known that solutions to the inviscid Burgers` equation form shock singularities in finite time, even when launched from smooth data. A far less documented fact, at least in the popular works on 1D hyperbolic conservation laws, is that shock singularities are intimately tied to a lack-of-uniqueness for the classical Burgers` equation.
We prove that, locally, solutions to the Compressible Euler equations do not suffer from the same lack-of-uniqueness, even though they can be written as a coupled system of Burgers` in isentropic plane-symmetry. Roughly, the saving grace is that Euler flow involves two speeds of propagation, and one of them ``prevents the mechanism driving the lack-of-uniqueness. Analytically, this is done by explicitly constructing a portion of the boundary of classical hyperbolic development for shock forming data. This boundary is a connected co-dimension 1 submanifold of Cartesian space, and we will discuss the delicate geo-analytic degeneracies and difficulties involved. This is joint work with Jared Speck. |
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