Abstract
It is evident from the lectures at this meeting that the subject of systems of hyperbolic conservation laws is flourishing as one of the prototypical examples of the modern mode of applied mathematics. Research in this area often involves strong and close interdisciplinary interactions among diverse areas of applied mathematics including
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(1)
Large (and small) scale computing
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(2)
Asymptotic modelling
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(3)
Qualitative modelling
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(4)
Rigorous proofs for suitable prototype problems
combined with careful attention to experimental data when possible. In fact, the subject is developing at such a rapid rate that new predictions of phenomena through a combination of theory and computations can be made in regimes which are not readily accessible to experimentalists. Pioneering examples of this type of interaction can be found in the papers of Grove, Glaz, and Colella in this volume as well as the recent work of Woodward, Artola, and the author ([1], [2], [3], [4], [5], [6]). In this last work, novel mechanisms of nonlinear instability in supersonic vortex sheets have been documented and explained very recently through a sophisticated combination of numerical experiments and mathematical theory.
partially supported by grants N.S.F. DMS 8702864, A.R.O. DAAL03-89-K-0013, O.N.R. N00014-89-J-1044
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References
M. Artola and A. Majda, Nonlinear development of instabilities in supersonic vortex sheets I: the basic kink modes, Physica 28D, pp. 253–281, 1988.
M. Artola and A. Majda, Nonlinear development of instabilities in supersonic vortex sheets II: resonant interaction among kink modes, (in press S.I.A.M. J. Appl. Math., to appear in 1989).
M. Artola and A. Majda, Nonlinear kink modes for supersonic vortex sheets, in press, Physics of Fluids A, to appear in 1989.
P. Woodward, in Numerical Methods for the Euler Equations of Fluid Dynamics, eds. Angrand, Dewieux, Desideri, and Glowinski, S.I.A.M. 1985.
P. Woodward, in Astrophysical Radiation Hydrodynamics, eds. K.H. Winkler and M. Norman, Reidel, 1986.
P. Woodward and K.H. Winkler, Simulation and visualization of fluid now in a numerical laboratory, preprint October 1988.
A. Majda, Vorticity and the mathematical theory of incompressible fluid flow, Comm. Pure Appl. Math 39, (1986), pp. S 187–220.
A. Majda, Mathematical Fluid Dynamics: The Interaction of Nonlinear Analysis and Modern Applied Math, Centennial Celebration of A.M.S., Providence, RI, August 1988 (to be published by A.M.S. in 1990).
R. Courant and K. Friedrichs, Supersonic Flow and Shock Waves, Springer-Verlag, New York, 1949.
C.S. Morawetz, The mathematical approach to the sonic barrier, Bull. Amer. Math. Soc., 6, #2 (1982), pp. 127–145.
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sciences 53, Springer-Verlag, New York 1984.
W. Hayes, The vorticity jump across a gas dynamic discontinuity, J. Fluid Mech. 2 (1957), pp. 595–600.
A. Majda and E. Thomann, Multi-dimensional shock fronts for second order wave equations, Comm. P.D.E., 12 (1987), pp. 777–828.
H. Glaz, P. Colella, I.I. Glass, and R. Deschambault, A detailed numerical, graphical, and experimental study of oblique shock wave reflections, Lawrence Berkeley Report, April 1985.
M. Van Dyke, An Album of Fluid Motion, Parabolic Press, Stanford, 1982.
C.S. Morawetz, On the non-existence of continuous transonic flows past profiles, I, II, III, Comm. Pure Appl. Math. 9, pp. 45–68, 1956
C.S. Morawetz, On the non-existence of continuous transonic flows past profiles, I, II, III, Comm. Pure Appl. Math. 10, pp. 107–132, 1957
C.S. Morawetz, On the non-existence of continuous transonic flows past profiles, I, II, III, Comm. Pure Appl. Math. 11, pp. 129–144, 1958.
J. Hunter and J.B. Keller, Weak shock diffraction, Wave Motion 6 (1984), pp. 79–89.
E. Harabetian, Diffraction of a weak shock by a wedge, Comm. Pure Appl. Math. 40 (1987), pp. 849–863.
D. Jones, P. Martin, and C. Thornhill, Proc. Roy. Soc. London A, 209, 1951, pp. 238–247.
J.B. Keller and A. A. Blank, Diffraction and reflection of pulses by wedges and corners, Comm. Pure Appl. Math. 4 (1951), pp. 75–94.
J. Hunter, Hyperbolic waves and nonlinear geometrical acoustics, in Proceedings of 6th Army Conference on Applied Mathematics and Computations, Boulder, CO, May 1988 (to appear).
D.G. Crighton, Basic theoretical nonlinear acoustics, in Frontiers in Physical Acoustics, Proc. Int. School of Physics Enrico Fermi, Course 93, North-Holland, Amsterdam (1986).
D.G. Crighton, Model equations for nonlinear acoustics, Ann. Rev. Fluid. Mech. 11 (1979), pp.11–33.
A. Majda, Nonlinear geometric optics for hyperbolic systems of conservation laws, in Oscillation Theory, Computation, and Methods of Compensated Compactness, IMA Volume 2, 115–165, Springer-Verlag, New York, 1986.
A. Majda and R. Rosales, Nonlinear mean field-high frequency wave interactions in the induction zone, S.I.A.M. J. Appl. Math., 47 (1987), pp. 1017–1039.
R. Almgren, A. Majda and R. Rosales, Rapid initiation through high frequency resonant nonlinear acoustics, (submitted to Combustion Sci. and Tech., July 1989).
R. Diperna and A. Majda, The validity of nonlinear geometric optics for weak solutions of conservation laws, Commun. Math. Physics 98 (1985), pp. 313–347.
J. K. Hunter and J.B. Keller, Weakly nonliner high frequency waves, Comm. Pure Appl. Math. 36 (1983), pp. 543–569.
J.K. Hunter, A. Majda, and R.R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves, II: several space variables, Stud. Appl. Math. 75 (1986), pp. 187–226.
A. Majda and R.R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves, I: a single space variable, Stud. Appl. Math. 71 (1984), pp. 149–179.
A. Majda, R. Rosales, M. Schonbek, A canonical system of integro-differential equations arising in resonant nonlinear acoustics, Studies Appl. Math, in 1989 (to appear).
R. Pego, Some explicit resonanting waves in weakly nonlinear gas dynamics, Stud. Appl. Math, in 1989 (to appear).
P. Cehelsky and R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves in the presence of shocks: A single space variable in a homogeneous time independent medium, Stud. Appl. Math. 74 (1986), pp. 117–138.
L. Tartar, Compensated compactness and applications to partial differential equations, in Research Notes in Mathematics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 4, R. Knops, ed., Pitman, London, 1979.
R. Diperna, Convergence of approximate solutions to conservation laws, Arch Rat. Mech. Anal. 82 (1983), pp. 27–70.
R. Diperna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), pp. 1–30.
M. Rascle and D. Serre, Comparite par compensation et systemes hyperboliques de lois de conservation, Applications C.R.A.S. 299 (1984), pp. 673–679.
D. Serre, La compucite par compensation pour les systems hyperboliques nonlineaires de deux equations a une dimension d’espace, J. Maths. Pures et Appl., 65 (1986), pp. 423–468.
W. Ficket and W. Davis, Detonation, Univ. California Press, Berkeley, 1979.
A. Majda, High Mach number combustion, in Reacting Flows: Combustion and Chemical Reactors, AMS Lectures in Applied Mathematics, 24, 1986, pp. 109–184.
A. Majda and V. Roytburd, Numerical study of the mechanisms for initiation of reacting shock waves, submitted to S.I.A.M. J. of Sci. and Stat. Computing in May 1989.
A. Majda, A qualitative model for dynamic combustion, S.I.A.M. J. Appl. Math., 41 (1981), pp. 70–93.
R. Rosales and A. Majda, Weakly nonlinear detonation waves, S.I.A.M. J. Appl. Math., 43 (1983), pp. 1086–1118.
A. Majda and V. Roytburd, Numerical modeling of the initiation of reacting shock waves, in Computational Fluid Mechanics and Reacting Gas Flows, B. Engquist et al eds., I.M.A. Volumes in Mathematics and Applications, Vol. 12, 1988, pp. 195–217.
A. Majda and R. Rosales, A theory for spontaneous Mach stem formation in reacting shock fronts, I: the basic perturbation analysis, S.I.A.M. J. Appl. Math. 43, (1983), pp. 1310–1334.
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Majda, A.J. (1991). One Perspective on Open Problems in Multi-Dimensional Conservation Laws. 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_18
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