Abstract
The subject of nonlinear hyperbolic waves is surveyed, with an emphasis on the discussion of a number of open problems.
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Supported in part by the Applied Mathematical Sciences Program of the DOE, grant DE-FG02-88ER25053
Supported in part by the NSF Grant DMS-8619856
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References
A. Aavatsmark, To Appear, “Capillary Energy and Entropy Condition for the Buckley-Leverett Equation” Contemporary Mathematics.
M. Artola and A. Majda, 1987, “Nonlinear Development of Instabilities in Supersonic Vortex Sheets” Physica D 28, pp. 253–281.
M. Artola and A. Majda, 1989, “Nonlinear Kink Modes for Supersonic Vortex Sheets,” Phys. Fluids.
J. B. Bell, J. A. Trangenstein, and G. R. Shubin, 1986, “Conservation Laws of Mixed Type Describing Three-Phase Flow in Porous Media” SIAM J. Appl. Math. 46, pp. 1000–1017.
G. Caginalp, 1986, “An Analysis of a Phase Field Model of a Free Boundary” Archive for Rational Mechanics and Analysis 92, pp. 205–245.
G. Caginalp, 1986, “The Role of Microscopic Anisotropy in the Macroscopic Behavior of a Phase Field Boundary” Ann. Phys. 172, pp. 136–146.
G. Caginalp, To Appear, Phase Field Models: Some Conjectures on Theorems for their Sharp Interface Limits
G. Caginalp, To Appear, Stefan and Hele-Shaw Type Models as Asymptotic Limits of the Phase Field Equations
G. Caginalp, To Appear, “The Dynamics of a Conserved Phase Field System: Stephan-like, Hele-Shaw and Cahn-Hilliard Models as Asymptotic Limits” IMA J. Applied Math.
Tung Chang and Ling Hsiao, 1988, The Riemann problem and Interaction of Waves in Gas Dynamics (John Wiley, New York).
Guiqiang Chen, 1987, “Overtaking of Shocks of the same kind in the Isentorpic Steady Supersonic Plane Flow” Acta Math. Sinica 7, pp. 311–327.
I-Liang Chern and T.-P. Liu, 1987, “Convergence to Diffusion Waves of Solutions for Viscous Conservation Laws” Comm. in Math. Phys. 110, pp. 503–517.
F. Furtado, 1989, “Stability of Nonlinear Waves for Conservation Laws” New York University Thesis.
F. Furtado, Eli Isaacson, D. Marchesin, and B. Plohr, To Appear, Stability of Riemann Solutions in the Large
X. Garaizar, 1989, “The Small Anisotropy Formulation of Elastic Deformation” Acta Applicandae Mathematica 14, pp. 259–268.
X. Garaizar, 1989, Private Communication
C. Gardner, J. Glimm, O. McBryan, R. Menikoff, D. H. Sharp, and Q. Zhang, 1988, “The Dynamics of Bubble Growth for Rayleigh-Taylor Unstable Interfaces,” Phys. of Fluids 31, pp. 447–465.
H. Gilquin, 1989, “Glimm’s scheme and conservation laws of mixed type” SIAM Jour. Sci. Stat. Computing 10, pp. 133–153.
J. Glimm, C. Klingenberg, O. McBryan, B. Plohr, D. Sharp, and S. Yaniv, 1985, “Front Tracking and Two Dimensional Riemann Problems” Advances in Appl. Math. 6, pp. 259–290.
J. Glimm and D.H. Sharp, 1986, “An S Matrix Theory for Classical Nonlinear Physics” Foundations of Physics 16, pp. 125–141.
J. Glimm and David H. Sharp, 1987, “Numerical Analysis and the Scientifíc Method” IBM J. Research and Development 31, pp. 169–177.
J. Glimm, 1988, “The Interactions of Nonlinear Hyperbolic Waves,” Comm. Pure Appl. Math. 41, pp. 569–590.
J. Glimm, Jan 1988, “The Continuous Structure of Discontinuities” in Proceedings of Nice Conference.
J. Glimm and X.L. Li, 1988, “On the Validation of the Sharp-Wheeler Bubble Merger Model from Experimental and Computational Data” Phys. of Fluids 31, pp. 2077–2085.
J. Glimm, X. L. Li, R. Menikoff, D. H. Sharp, and Q. Zhang, To appear, A Numerical Study of Bubble Interactions in Rayleigh-Taylor Instability for Compressible Fluids
J. Glimm, To appear, “Scientific Computing: von Neumann’s vision, today’s realities and the promise of the future” in The Legacy of John von Neumann, ed. J. Impagliazzo (Amer. Math. Soc, Providence).
J. Goodman and X. Xin, To Appear, Viscous Limits for Piecewise Smooth Solutions to Systems of Conservation Laws
J. W. Grove and R. Menikoff, 1988, “The Anomalous Reflection of a Shock Wave through a Material Interface” in preparation.
L. F. Henderson, 1988, “On the Refraction of Longitudinal Waves in Compressible Media,” LLNL Report UCRL-53853.
D. Hoff and T.-P. Liu, To Appear, “The Inviscid Limit for the Navier-Stokes equations of Compressible, Isentropic now with shock data” Indiana J. Math..
H. Holden, 1987, “On the Riemann Problem for a Prototype of a Mixed Type Conservation Law” Comm. Pure Appl. Math. 40, pp. 229–264.
H. Holden and L. Holden, To Appear, “On the Riemann problem for a Prototype of a Mixed Type Conservation Law II” Contemporary Mathematics.
Ling Hsiao and Tung Chang, 1980 Acta Appl. Math. Sinica 4, pp. 343–375.
P.-T. Kan, 1989, “On the Cauchy Problem of a 2 × 2 System of Nonstrictly Hyperbolic Conservation Laws,” NYU Thesis.
B. Keyfitz, To Appear, “Criterion for Certain Wave Structures in Systems that Change Type” Contemporary Mathematics.
T.-P. Liu, 1985, “Nonlinear stability of shock waves for viscous conservation laws,” Memoir, AMS:328, pp. 1–108.
T.-P. Liu, 1987, “Hyperbolic Conservation Laws with Relaxation” Comm Math Phys 108, pp. 153–175.
T.-P. Liu and X. Xin, To Appear, Stability of Viscous Shock Wave Asociated with a System of Nonstrictly Hyperbolic Conservation Laws
A. Majda and V. Roytburd, To Appear, “Numerical Study of the Mechanisms for Initiation of Reacting Shock Waves,” Siam J. Sci Stat Comp.
R. Menikoff and B. Plohr, 1989, “Riemann Problem for Fluid Flow of Real Materials” Rev. Mod. Phys. 61, pp. 75–130.
R. Menikoff, 1989, Private Communication
R. von Mises, 1958, Mathematical Theory of Compressible Fluid Flow (Academic Press, New York).
R. L. Rabie, G. R. Fowles, and W. Fickett, 1979, “The Polymorphic Detonation,” Phys. of Fluids 22, pp. 422–435.
K. I. Read, 1984, “Experimental Investigation of Turbulent Mixing by Rayleigh-Taylor Instability” Physica 12D, pp. 45–48.
D. H. Sharp and J. A. Wheeler, 1961, “Late Stage of Rayleigh-Taylor Instability” Institute for Defense Analyses.
M. Shearer, 1987, “Loss of Strict Hyperbolicity in the Buckley-Leverett Equations of Three Phase Flow in a Porous Medium.” in Numerical Simulation in Oil Recovery, ed. M. Wheeler (Springer Verlag, New York).
Z. Tang and T. C. T. Ting, 1987, “Wave Curves for the Riemann Problem of Plane Waves in Simple Isotropic Elastic Solids” Int. J. Eng. Science 25, pp. 1343–1381.
P. Woodward, 1985, “Simulation of the Kelvin-Helmholtz Instability of a Supersonic Slipsuface with a Piecewise Parabolic Method” Proc. INRIA Workshop on Numerical Methods for Euler Equations, p. 114.
J. A. Zufiria, “Vortex-in-Cell Simulation of Bubble Competition in Rayleigh-Taylor Instability,” Preprint, 1988.
J. A. Zufiria, 1988, “Bubble Competition in Rayleigh-Taylor Instability” Phys. of Fluids 31, pp. 440–446.
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Glimm, J. (1991). Nonlinear Waves: Overview and Problems. In: Glimm, J., Majda, A.J. (eds) Multidimensional Hyperbolic Problems and Computations. The IMA Volumes in Mathematics and Its Applications, vol 29. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9121-0_8
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