Simple waves and double waves as exact solutions of quasilinear hyperbolic equations in hydrodynamics may blow up, and, so, generally, there are points at which these solutions are violated at some definite instants of time. Finding the first instant of time for the violation of a simple wave solution is a known method, while this problem for the double waves is not a well defined and an easy task. Here, a two and a three dimensional double wave solutions are reviewed for some hydrodynamical systems to show that, for the 2-D case for some special solutions, one can find a local minimum for the critical time, while for some other solutions not. For the 3-D case, generically, there is no local minimum for the critical time, but at very special situations, the local minimum critical times may exist. Moreover, the dynamics of critical curves and points are briefly discussed.
References
1.
R.
Von Mises
, Mathematical Theory of Compressible Flow
(Academic Press
, New York
, 1958
).2.
M.
Burnat
, “Cauchy's problem for the compressible flow of simple wave type (Compressible flow of simple wave type in gas dynamics, solving Cauchy problem)
,” in Proceedings of the 5th Symposium of Fluid Dynamics Transactions
, Jablonna, Poland, 26 Aug–2 Sept 1961 (1964
), pp. 31
–57
.3.
M.
Burnat
, “Theory of simple waves for nonlinear systems of partial differential equations and applications to gas dynamics
,” Arch. Mech. Stos.
18
(4
) (1966
).4.
R.
Courant
and K. O.
Friedrichs
, Supersonic Flow and Shock Waves
(Springer-Verlag
, New York
, 1976
).5.
6.
A. J.
Chorin
and J. E.
Marsden
, A Mathematical Introduction to Fluid Mechanics
(Springer-Verlag
, New York
, 1979
).7.
B. L.
Rozdestvenskii
and N. N.
Janenko
, Systems of Quasilinear Equations and Their Applications to Gas Dynamics
, Translations of Mathematical Monographs Vol. 55 (American Mathematical Society
, Providence, RI
, 1980
).8.
9.
L. D.
Landau
and E. M.
Lifshitz
, Electrodynamics of Continuous Media
, 2nd ed. (Pergamon
, Oxford
, 1984
).10.
A. I.
Akhiezer
, I. A.
Akhiezer
, R. V.
Polovin
, A. G.
Sitenko
, and K. N.
Stepanov
, Plasma Electrodynamics
(Pergamon
, Oxford
, 1975
).11.
12.
J.
Smoller
, Shock Waves and Reaction Diffusion Equations
(Springer-Verlag
, Berlin
, 1983
).13.
J.
Goodman
, “Nonlinear asymptotic stability of viscous shock profiles for conservation laws
,” Arch. Ration. Mech. Anal.
95
, 325
–344
(1986
).14.
F.
John
, Nonlinear Wave Equations, Formation of Singularities
, Pitcher's Lectures in the Mathematical Sciences held at Lehigh University April 1989 (American Mathematical Society
, Providence
, 1990
).15.
C. M.
Dafermos
, Hyperbolic Conservation Laws in Continuum Physics
, 2nd ed. (Springer-Verlag
, Berlin
, 2005
).16.
L.
Caffarelli
and A.
Vasseur
, “Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation
,” Ann. Math.
171
(3), 1903
–1930
(2010
).17.
M.
Burnat
, “Geometrical methods in fluid mechanics
,” Fluid Dyn. Trans.
6
, 115
–186
(1967
).18.
19.
M.
Burnat
, “The method of Riemann invariants for multi-dimensional nonelliptic systems
,” Bull. Acad. Pol. Sci., Ser. Sci. Technol.
17
, 1019
–1026
(1969
).20.
M.
Burnat
, “The method of characteristics and Riemann invariants for multidimensional hyperbolic systems
,” Siberian Math. J.
11
, 210
–232
(1970
).21.
G.
Boillat
, “Simple waves in N-dimensional propagation
,” J. Math. Phys.
11
, 1482
–1483
(1970
).22.
G. M.
Webb
, R.
Ratkiewicz
, M.
Brio
, and G. P.
Zank
, “Multidimensional simple waves in gas dynamics
,” J. Plasma Phys.
59
, 417
–460
(1998
).23.
T.
Sahihi
, H.
Eshraghi
, A.
Mahdipour-Shirayeh
, and P. K.
Shukla
, “Multi-dimensional simple waves in fully relativistic fluids
,” J. Phys. A: Math. Theor.
43
, 165501
(2010
).24.
M.
Burnat
, “The method of solution of hyperbolic systems by means of combining simple waves
,” Fluid Dyn. Trans.
3
, 23
–40
(1967
).25.
A. F.
Sidorov
and N. N.
Yanenko
, “On the problem of nonstationary plane fluxes of polytropic gases with straight line characteristics (in Russian)
,” Dokl. Akad. Nauk SSSR
123
, 832
–834
(1958
).26.
L. V.
Komarovskii
, “An accurate solution of the three-dimensional equations for a nonsteady gas-flow of the double wave type
,” Sov. Phys. Dokl.
135
, 1163
–1165
(1960
).27.
M.
Burnat
, “The hyperbolic double waves
,” Bul. Acad. Poi. Sci., Ser. Sci.
17
(10
), 97
–106
(1969
).28.
Z.
Peradzynski
, “Nonlinear plane k-waves and Riemann invariants
,” Bull. Acad. Pol. Sci., Ser. Sci. Technol.
19
, 625
–632
(1971
).29.
Z.
Peradzynski
, “Riemann invariants for the nonplanar k-waves
,” Bull. Acad. Pol. Sci., Ser. Sci. Technol.
19
, 717
–724
(1971
).30.
W.
Zajaczkowski
, “Riemann invariants interaction in MHD double waves
,” Demonstr. Math.
12
, 543
–563
(1979
).31.
D. D.
Tskhakaya
and H.
Eshraghi
, “Two-dimensional double simple waves in a pair plasma at relativistic temperatures
,” Phys. Plasmas
9
, 2518
–2525
(2002
).32.
D. D.
Tskhakaya
and H.
Eshraghi
, “On the theory of magneto-sound double simple waves
,” J. Plasma Phys.
74
, 455
–471
(2008
).© 2016 Author(s).
2016
Author(s)
You do not currently have access to this content.
