arXiv subjects

Joseph Roberts

Publications and source records attributed to Joseph Roberts.

3 recordsLinked to original sources

arXiv2013

Steady and self-similar solutions of non-strictly hyperbolic systems of conservation laws

We consider solutions of two-dimensional $m \times m$ systems hyperbolic conservation laws that are constant in time and along rays starting at the origin. The solutions are assumed to be small $L^\infty$ perturbations of a constant state and entropy admissible, and the system is assumed to be non-strictly hyperbolic with eigenvalues of constant multiplicity. We show that such a solution, initially assumed bounded, must be a special function of bounded variation, and we determine the possible configuration of waves. As a corollary, we extend some regularity and uniqueness results for some one-dimensional Riemann problems.

math.AP↗

arXiv2012

Steady and self similar full Euler flow

We consider solutions to the full (non-isentropic) two-dimensional Euler equations that are constant in time and along rays emanating from the origin. We prove that for a polytropic equation of state, entropy admissible solutions in $L^\infty$ with non-vanishing velocity, density, and internal energy must be $BV$. Moreover, we obtain some results concerning the structure of such solutions.

math.AP↗

arXiv2012

Steady self-similar inviscid flow

We consider solutions of the 2-d compressible Euler equations that are steady and self-similar. They arise naturally at interaction points in genuinely multi-dimensional flow. We characterize the possible solutions in the class of flows L^\infty-close to a constant supersonic background. As a special case we prove that solutions of 1-d Riemann problems are unique in the class of small L^\infty functions. We also show that solutions of the backward-in-time Riemann problem are necessarily BV.

math.AP↗