R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 92
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15.2). Here v . Mises' conception differsbasically from that of R . COURANT and K . O. FRIEDRICHS [21], p. 126, and others,who parallel contact discontinuities with shocks. For v . Mises the contact discontinuities of compressible flow are discontinuities across characteristics, satisfying thecondition (10), for which the variable itself undergoes an abrupt change; whereas ashock is strictly speaking not a phenomenon of inviscid nonconducting fluid flowtheory. (See Sees. 14.1, 14.2, 22.1, and 22.2.)Article 1039.
A system of this form is often called quasilinear (see also N o t e 27).40. Here, where η = k = 2, our results can be obtained in terms of the theory oftwo linear algebraic equations with two unknowns. An elegant presentation alongthese lines is given in SAUER, [7], p. 63 ff. T h e complete and symmetric compatibilityrelations (4a)-(4d) and (5) are new.41. This theorem, formulated in a mathematically more rigorous way, has beenproved by H .
L E W Y , " U b e r das Anfangswertproblem bei einer hyperbolischen nichtlinearen partiellen Differentialgleichung zweiter Ordnung mit zwei unabhangigenVeranderlichen", Math. Ann. 98 (1927), pp. 179-191. A proof is in COURANT-FRIEDRICHS [21], p.
48 ff. W e also refer the reader again to T . L E V I - C I V I T A [6], to F. TRICOMI,cit. N o t e 27, and t o R. SAUER [7].42. This reasoning, which at the same time provides a numerical method, goesback to J . M A S S A U , Memoiresur Vintegrationgraphiquedes equationsaux deriveespartielles, Gand: van Goethem, 1900 (reviewed in Enzykl. math. Wiss. II/3 (1915),p. 162, article by C. Runge and F. A . W i l l e r s ) , reprinted as Edition du Centenaire,Mons, 1952.43. W e have seen that Massau's method (see N o t e 42) fails if any cross line assumes characteristic direction. W e must, however, beware of the mistaken belief thatif this method encounters no obstacle—for what seems a sufficiently small mesh—it necessarily yields approximate knowledge of the desired solution.
In fact, considerthe system v (du/dx)= u (dv/dy),v (du/dy)= u (dv/dx),whose characteristicsare the straight =h45°-lines. I t admits the particular solution u= [1 + 2 ( x -f y )\~ ,ν = [1 -f 4:n/] , where υ —> «> on the hyperbola xy — — Y±. Along the noncharacteristicsegment from (0,0) to (2,0), say, this solution takes on the regular boundary valuesu = (1 + 2x )~ , ν = 1. Massau's method applied (for a chosen mesh) to the characteristic triangle (0,0), (2,0), ( 1 , - 1 ) does not encounter any difficulty (all cross linesmay be taken horizontal thus having nowhere characteristic direction) and leads towell-defined finite values which give no indication of the singularity in the exactsolution.
On the other hand, in a sufficiently close neighborhood of the segment (0,0),(0,2) (whose distance from the x-axis is less than 1 — Λ/3/2) the exact solution iseverywhere finite and Massau's method gives an approximation to it. (This examplewas communicated to H . Geiringer by M .
Schiffer.) W e do not wish to imply thatsuch a situation will arise in the fluid-dynamical case, though the converse has neverbeen demonstrated.22222_12x2l474NOTES A N D A D D E N D AArticle 1044. W e may wonder whether by imposing suitable restrictions on the coefficientsof the system (1) we could exclude cases such as the example in N o t e 43. T h e answer is negative, since for a nonlinear differential equation the singularities of solutions are not determined by singularities of the coefficients. F o r example, for the nonlinear ordinary equation dy/dx = y , the general solution y — (a — x)~ has a poleat an arbitrary point χ = a, which can in no way be predicted from the coefficientsof the given equation.2l45.
I t is a frequent mistake t o assume that in the characteristic boundary-valueproblem a solution is guaranteed in a small neighborhood of AB and of AC. I t isguaranteed only in a neighborhood of the point A. (Counter-examples can be constructed in various ways.) This can be understood intuitively from a comparisonof Figs. 45 and 46. I n F i g . 45 the whole row of points A'B'adjacent t o A Βis derived directly from the given data along AB. I n F g . 46, however, only the position of Ρ4 follows directly from the given data.
F o r all oxher points, say those adjacentto AB we need in addition t o the given data the derived values at P , etc., and hencea certain uniformity concerning these derived values. A correct mathematical existence proof for this boundary-value problem is due t o H . L e w y , see N o t e 41, andCOURANT-HILBERT [2], where the problem is reduced to one of a system of ordinarydifferential equations.;}446. For the ' ' m i x e d ' ' boundary-value problem where we know compatible valuesof u and ν along the characteristic AC and one variable along ΑΑι , existence can beproved in the neighborhood of A only. I n relation t o these more general boundaryvalue problems see papers b y : H . BECKERT, " U b e r quasilineare hyperbolise heSysteme partieller Differentialgleichungen erster Ordnung mit zwei unabhangigenVariablen.
Das Anfangswertproblem, die gemischte Randwertaufgabe, das charakteristische P r o b l e m " , Ber. Verhandl.sdchs. Akad.Wiss.Leipzig,Math.-Naturw.KI.97 (1950), p. 68 ff.; W . H A A C K and G. H E L L W I G , " U b e r Systeme hyperbolischer Differentialgleichungen erster Ordnung. I " , Math. Z. 53 (1950), pp. 244-266; I I , ibid. p p .340-356; and R .
COURANT and P. L A X , " O n nonlinear partial differential equations forfunctions of t w o independent v a r i a b l e s " , Communs. Pure Appl. Math. 2 (1949), p p .255-273.47. Riemann developed this method in the paper quoted in N o t e 23 as a sort ofappendix t o the physical theory contained therein. H e considers an equation suchas (12.43) except that 2/(£ + η) is replaced by — m, a function of (£ + η). His methodis completely explained by means of this example. T h e first t o consider in detail thegeneral equation (11) was G. DARBOUX, in his Legons sur la theorie generate des surfaces, 2nd ed., V o l . I I , P a r i s : Gauthier-Villars, 1915, p.
71 ff. (1st ed., 1888).48. Formula (17) is called Riemann's formula. Riemann's method has been generalized b y various mathematicians, above all b y J . HADAMARD, cit. N o t e 1.24, whodeveloped an integration theory for the general second-order linear equation in nindependent variables. See the presentation in SAUER, [7], p.
194 ff.; cf. also BERGMAN-SQUIFFER, [1], p. 365 ff., and the paper b y M . Riesz quoted in N o t e 1.24.49. Regarding the determination of the function Ω, Riemann adds: " T h e determination of such a solution (our Ω ) is often made possible by the consideration of aparticular case . . . " . H . Weber, the editor of Riemann's works, explains this remarkas follows: Since the determination of Ω is independent of the particular boundaryvalues given for U, we may try t o find a particular solution U for conveniently chosenvalues of U and its derivatives on a conveniently chosen 6 ; then the Riemann formula(17) gives Ω. T h i s simple and v e r y suggestive idea is carried out for Riemann's equation ( N o t e 47).50.
F o r examples of Riemann functions see Sec. 12. 4, and N o t e 111.22.CHAPTER475IIIArticle 1051. A t this stage, our point of view is that, if all variables are considered as functions of u,v, then x(u,v)by / ^ 0] u(x,y)and y(u,v)and v{x,y)must satisfy (22) if after inversion [guaranteedare to satisfy ( 1 ) . One does not obtain all solutions of (1)in this w a y : those for which j = 1/J vanishes are " l o s t " (see A r t . 18).
If the two transitions, the one from the z,2/-plane to the w,^-plane and the reverse one are consideredseparately we see that j 9* 0 is the condition for the first, J ^ 0 that for the secondone. M o r e will be found in Arts. 17, 18, a n d 19.CHAPTER IIIArticle 111. See N o t e 1.13.2. A complete qualitative discussion of these flows has been given by G .
S. S.LUDFORD, " T h e classification of one-dimensional flows and the general shock problem of a compressible, viscous, heat-conducting fluid", / . Aeronaut. Sci. 18 (1951),pp. 830-834. For the special case Ρ = % (cf. E q . (32)), the equations can be integratedexplicitly, see M . MORDUCHOW and P. A . L I B B Y , " O n a complete solution of the onedimensional flow equations of a viscous, heat-conducting, compressible g a s " , / .Aeronaut.
Sci. 16 (1949), pp. 674-684.3. G . I . T A Y L O R , " T h e conditions necessary for discontinuous motion in gases",Proc. Roy. Soc. A84 (1910), pp. 371-377. This paper appears in [20] and is essentiallyreproduced in V o l . I l l of [23] in the article by G .
I . T A Y L O R and J. W . MACCOLL, " T h emechanics of compressible fluids", pp. 209-250. T a y l o r also considered the case μ = 0,which had been previously discussed by W . J. M . R A N K I N E , " O n the thermodynamictheory of waves of finite longitudinal disturbance", Phil. Trans. Roy. Soc. London160 (1870), pp. 277-286. Similar results to T a y l o r ' s were obtained by LORD R A Y L E I G H ," A e r i a l plane waves of finite a m p l i t u d e " , Proc. Roy. Soc.
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