On the formation of singularities for the compressible Euler equations - Bayles-Rea, T (2022)
The purpose of this thesis is to study the phenomenon of singularity formation in large data problems for C^1 solutions to the Cauchy problem of the compressible Euler equations. The classical theory established by P. D. Lax in 1964 (J. Math. Phys. 5: 611-614) shows that, for 2x2 hyperbolic systems, the break-down of C^1 solutions occurs in finite time if initial data contain any compression in some truly nonlinear characteristic field under some additional conditions, which include genuine nonlinearity and the strict positivity of the difference between two corresponding eigenvalues. These strong structural assumptions mean that it is highly non-trivial to apply this theory to archetypal systems of conservation laws, such as the (1+1)-dimensional relativistic Euler equations.