| Summary: | <p>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.</p>
<p>Actually, in the (1+1)–dimensional spacetime setting, if the mass-energy density rho does not vanish initially at any finite point, the essential difficulty in considering the possible break-down is coming up with a way to obtain sharp enough control on the lower bound for rho.</p>
<p>In Chapter 2 and Chapter 3, we consider the (1+1)–dimensional isentropic Euler equations in both classical and relativistic settings. We will establish a sufficiently sharp lower bound of rho, and show Lax's theory indeed holds for these systems. Furthermore, we will give a better lower bound of rho, at the expense of restricting the class of initial data further.</p>
<p>In Chapter 4 and Chapter 5, we consider the (1+1)–dimensional non-isentropic Euler equations in both classical and relativistic settings. As the non-isentropic Euler equations are 3x3 systems, Lax's theory no longer directly applies. Instead, we will establish a sufficiently sharp lower bound of rho, and give a sufficient condition for the solution to break down in finite time.</p>
<p>In Chapter 6 and Chapter 7, we consider the multidimensional isentropic Euler equations with spherical symmetry in both classical and relativistic settings. With spherical symmetry, the system will have a singularity at the origin. To bypass this, we consider a specific set of initial data such that the local existence and uniqueness result can be established. Same as in the non-isentropic case, we will give a sufficient condition for the solution to break down in finite time.</p>
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