References
- Inagaki F, Kuypers MMM, Tsunogai U, et al. Microbial community in a sediment-hosted CO2 lake of the southern Okinawa Trough hydrothermal system. Proc Natl Acad Sci USA. 2006;103:14164–14169. doi: 10.1073/pnas.0606083103
- House KZ, Schrag DP, Harvey CF, et al. Permanent carbon dioxide storage in deep-sea sediments. Proc Natl Acad Sci USA. 2006;103:12291–12295. doi: 10.1073/pnas.0605318103
- Span R, Wagner W. A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa. J Phys Chem Ref Data. 1996;25:1509–1596. doi: 10.1063/1.555991
- Hove J, Haugan PM. Dynamics of a CO2-seawater interface in the deep ocean. J Marine Res. 2005;63:563–577. doi: 10.1357/0022240054307867
- Yamane K, Aya I, Namie S, et al. Strength of CO2 hydrate membrane in sea water at 40 MPa. Ann NY Acad Sci. 2000;912:254–260. doi: 10.1111/j.1749-6632.2000.tb06779.x
- Craig W, Guyenne P, Kalisch H. Hamiltonian long-wave expansions for free surfaces and interfaces. Commun Pure Appl Math. 2005;58:1587–1641. doi: 10.1002/cpa.20098
- Kalisch H. Derivation and comparison of model equations for interfacial capillary-gravity waves in deep water. Math Comput Simul. 2007;74:168–178. doi: 10.1016/j.matcom.2006.10.008
- Glimm J. Solutions in large for nonlinear systems of equations. Commun Pure Appl Math. 1965;18:697–715. doi: 10.1002/cpa.3160180408
- Lax PD. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM CBMS-NSF Regional Conference Series in Applied Mathematics; 1973. p. 11.
- Liu TP. The deterministic version of the Glimm scheme. Commun Math Phys. 1977;57:135–148. doi: 10.1007/BF01625772
- Shao ZQ. Lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small BV initial data: shocks and contact discontinuities. J Math Anal Appl. 2012;387:698–720. doi: 10.1016/j.jmaa.2011.09.037
- Omelyanov GA. About the stability problem for strictly hyperbolic systems of conservation laws. Rend Sem Mat Univ Politec Torino. 2011;69:377–392.
- Stewart HB, Wendroff B. Two-phase flow: models and methods. J Comput Phys. 1984;56:363–409. doi: 10.1016/0021-9991(84)90103-7
- Kearsley AJ, Reiff AM. Existence of weak solutions to a class of nonstrictly hyperbolic conservation laws with non-interacting waves. Pacific J Math. 2002;205(1):153–170. doi: 10.2140/pjm.2002.205.153
- Schaeffer DG, Shearer M. The classification of 2×2 systems of non-strictly hyperbolic conservation laws, with application to oil recovery. Commun Pure Appl Math. 1987;40:141–178. doi: 10.1002/cpa.3160400202
- Shearer M. Loss of strict hyperbolicity in the Buckley–Leverett equations of three-phase flow in a porous medium. In: Wheeler M, editor. Numerical simulation in oil recovery. New York, Berlin, Heidelberg: Springer Verlag; 1988. p. 263–283.
- Kalisch H, Mitrovic D. Singular solutions of a fully nonlinear 2×2 system of conservation laws. Proc Edinb Math Soc. 2012;55:711–729. doi: 10.1017/S0013091512000065
- Kalisch H, Mitrovic D. Singular solutions for the shallow-water equations. IMA J Appl Math. 2012;77:340–350. doi: 10.1093/imamat/hxs014
- Kalisch H, Mitrovic D, Teyekpiti V. Delta shock waves in shallow water flow. Phys Lett A. 2017;381:1138–1144. doi: 10.1016/j.physleta.2017.02.007
- Hayes BT, LeFloch PG. Measure solutions to a strictly hyperbolic system of conservation laws. Nonlinearity. 1996;9(6):1547–1563. doi: 10.1088/0951-7715/9/6/009
- Danilov VG, Mitrović D. Delta shock wave formation in the case of triangular system of conservation laws. J Differ Equ. 2008;245:3704–3734. doi: 10.1016/j.jde.2008.03.006
- Korchinski C. Solution of a Riemann problem for a 2×2 system of conservation laws possessing no classical weak solution [PhD thesis]. Adelphi University; 1977.
- Tan DC, Zhang T, Zheng YX. Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws. J Differ Equ. 1994;112:1–32. doi: 10.1006/jdeq.1994.1093
- Choudhury AP. Singular solutions for 2×2 systems in nonconservative form with incomplete set of eigenvectors. Electron J Differ Equ. 2013;58:11.
- Huang F. Existence and uniqueness of discontinuous solutions for a hyperbolic system. Proc R Soc Edinb. 1997;127:1193–1205. doi: 10.1017/S0308210500027013
- Ercole G. Delta-shock waves as self-similar viscosity limits. Quart Appl Math. 2000;58:177–199. doi: 10.1090/qam/1739044
- Shen C. The Riemann problem for the pressureless Euler system with the Coulomb-like friction term. IMA J Appl Math. 2015;81:76–99.
- Sun M. Interactions of delta shock waves for the chromatography equations. Appl Math Lett. 2013;26:631–637. doi: 10.1016/j.aml.2013.01.002
- Wang G. One-dimensional nonlinear chromatography system and delta-shock waves. Z Angew Math Mech. 2013;64:1451–1469.
- Nedeljkov M. Shadow waves: entropies and interactions for delta and singular shocks. Arch Ration Mech Anal. 2010;197:489–537. doi: 10.1007/s00205-009-0281-2
- Nedeljkov M, Oberguggenberger M. Interactions of delta shock waves in a strictly hyperbolic system of conservation laws. J Math Anal Appl. 2008;344:1143–1157. doi: 10.1016/j.jmaa.2008.03.040
- LeVeque RJ. Numerical methods for conservation laws. Lectures in mathematics. Berlin: Birkhauser; 1992.
- Albeverio S, Shelkovich VM, On the delta-shock front problem. Analytical approaches to multidimensional balance laws. New York (NY): Nova Science Publishers; 2005. p.45–87.
- Danilov VG, Omel'yanov GA, Shelkovich VM. Weak asymptotic methods and interaction of nonlinear waves. In: Karasev MV, editor. Asymptotic methods for wave and quantum problems. American mathematical translations series 2, Vol. 208. Providence (RI): American Mathematical Society; 2003. p. 33–165.
- Danilov VG, Shelkovich VM. Dynamics of propagation and interaction of δ-shock waves in conservation law system. J Differ Equ. 2005;211:333–381. doi: 10.1016/j.jde.2004.12.011
- Kalisch H, Mitrovic D, Nordbotten JM. Non-standard shocks in the Buckley–Leverett equation. J Math Anal Appl. 2015;428:882–895. doi: 10.1016/j.jmaa.2015.03.041
- Rudin W. Functional analysis. 2nd ed. Singapore: McGraw-Hill; 1991.
- Danilov VG, Shelkovich VM. Delta-shock wave type solution of hyperbolic systems of conservation laws. Quart Appl Math. 2005;63:401–427. doi: 10.1090/S0033-569X-05-00961-8
- Keyfitz B, Kranzer H. Spaces of weighted measures for conservation laws with singular shock solutions. J Differ Equ. 1995;118:420–451. doi: 10.1006/jdeq.1995.1080