Journal of Hyperbolic Differential Equations

We study properties of evolution equations which are first order in time and arbitrary order in space (FTNS). Following Gundlach and Martín-García (2006) we define strong and symmetric hyperbolicity for FTNS systems and examine the relationship between these definitions, and the analogous concepts for first-order systems. We demonstrate equivalence of the FTNS definition of strong hyperbolicity with the existence of a strongly hyperbolic first-order reduction. We also demonstrate equivalence of the FTNS definition, up to N = 4, of symmetric hyperbolicity with the existence of a symmetric-hyperbolic first-order reduction.

We consider gas flow in pipe networks governed by the isothermal Euler equations. A set of coupling conditions is required to completely specify the Riemann problem at the junction. The momentum-related condition has no obvious expression and different approaches have been used in previous work. For the condition of equal momentum flux, Colombo and Garavello [A well posed Riemann problem for the p-system at a junction, Netw. Heterog. Media 1 (2006) 495–511] proved existence and uniqueness of solutions globally in time and locally in the subsonic region of the state space. If the entropy constraint is not considered, we are able to prove existence and uniqueness globally in the subsonic region for any momentum-related coupling constant satisfying a monotonicity requirement. The previously suggested conditions of equal pressure and equal momentum flux satisfy this requirement, but in general they both fail to fulfill the entropy constraint. The classical Bernoulli invariant is a natural scalar formulation of momentum conservation under ideal flow conditions. Our analysis shows that this invariant is monotone and unconditionally leads to solutions satisfying the entropy constraint. Of the coupling constants considered, this is therefore the only choice that guarantees the unique existence of entropy solutions to the N-junction Riemann problem for all initial data in the subsonic region.

Solutions of initial-boundary value problems for systems of conservation laws depend on the underlying viscous mechanism, namely different viscosity operators lead to different limit solutions. Standard numerical schemes for approximating conservation laws do not take into account this fact and converge to solutions that are not necessarily physically relevant. We design numerical schemes that incorporate explicit information about the underlying viscosity mechanism and approximate the physical-viscosity solution. Numerical experiments illustrating the robust performance of these schemes are presented.

We study small perturbations of the Friedman–Lemaître–Robertson–Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the space-like Cauchy hypersurfaces are diffeomorphic to 𝕋³. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. In our analysis, we construct harmonic-type coordinates such that the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends those of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715; C. Lübbe and J. A. Valiente Kroon, A conformal approach for the analysis of the nonlinear stability of pure radiation cosmologies, Ann. Phys. 328 (2013) 1–25], where analogous results were proved for the Euler–Einstein system under the equations of state , . The dust-Einstein system is the case c_s = 0. The main difficulty that we overcome here is that the dust's energy density loses one degree of differentiability compared to the cases , which introduces many obstacles for closing the estimates. To resolve this difficulty, we commute the equations with a well-chosen differential operator and derive elliptic estimates that complement the energy estimates of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715]. Our results apply in particular to small perturbations of the vanishing dust state containing vacuum regions.

This paper deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [Hyperbolic systems of balance laws with weak dissipation, J. Hyp. Diff. Equations 3 (2006) 505–527.], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented by [Ch. Arvanitis, Ch. Makridakis and A. E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic conservation laws, SIAM J. Numer. Anal. 42(4) (2004) 1357–1393]; [S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235–277] for systems of hyperbolic conservation laws without source terms.