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Open Journal of Applied Sciences, 2013, 3, 79-83 doi:10.4236/ojapps.2013.31B1016 Published Online April 2013 (http://www.scirp.org/journal/ojapps) Courant-Friedrichs’ Hypothesis and Stability of the Weak Shock Wave Satisfying the Lopatinski Condition Dmitry Tkachev, Alexander Blokhin Mechanics and Mathematics Department, Novosibirsk State University, Sobolev Institute of Mathematics, Novosibirsk, Russia Email: tkachev@math.nsc.ru, blokhin@math.nsc.ru Received 2013 ABSTRACT We are studying the problem of a stationary supersonic flow of an inviscid non-heat-conducting gas in thermodynami- cal equilibrium onto a planar infinite wedge. It is known that theoretically this problem has two solutions: the solution with a strong shock wave (when the velocity behind the front of the shock wave is subsonic) and the solution with a weak shock wave (when, generally speaking, the velocity behind the front of the shock wave is supersonic). In the pre- sent paper, the case of a weak shock wave is studied. It is proved that if the Lopatinski condition for the shock wave is satisfied (in a weak sense), then the corresponding linearized initial boundary-value problem is well-posed, and its clas- sical solution is found. In this case, unlike the case when the uniform Lopatinski condition holds, additional plane waves appear. It is shown that for compactly supported initial data the solution of the linearized problem converges in finite time to the zero solution. Therefore, for the case of a weak shock wave and when the Lopatinski condition holds in a weak sense these results complete the verification of the well-known Courant-Friedrichs' conjecture that the strong shock wave solution is unstable whereas the weak shock wave solution is stable. Keywords: Weak Shock Wave; Asymptotic Stability (in the Sense of Lyapunov) 1. Introduction It is well-known that the classical problem of a stationary supersonic flow of an inviscid non-heat-conducting gas onto an infinite plane wedge (with a sufficiently small angle σ at the vertex) has two solutions: one of them corresponds to the case of a strong shock wave, when the components of the velocity vector behind the shock satisfies the inequality (0 c is the sound speed of the gas behind the shock wave), and the second corresponds to the case of a weak shock wave, that is [1-3] (see Figure 1). 000 :,Uuv 22 2 00 0 uvc 22 00 0 uvc 2 In Figure 1 the angular coordinates determine strong and weak shocks respectively. The vector of flow velocity is parallel to the axis. Paradoxically, but in practice, in physical or numerical experiments, only the weak shock wave solution is actu- ally realized. One possible explanation was suggested by Courant and Friedrichs [1]. They conjectured that the solution corresponding to a strong shock wave is unsta- ble by Lyapunov whereas the weak shock wave solution is stable. The Courant-Friedrichs' hypothesis was verified in [4-7] (but this conclusion was based only on some quali- tative reasons). A strict mathematical justification (and this is very important) of this statement for the linearized problem appeared in recent years in [8-12]. Briefly speaking, it was shown in [8-12] that in the case of a strong shock wave (for compactly supported initial data!) the perturbation arrives to the wedge's vertex as time increases having the growth r (0 ) or a logarith- mic growth in space variables, and this causes instability of the steady-state solution under consideration. The situation with the weak shock wave is totally dif- ferent. In this case the perturbation decays with time. Moreover, this solution is asymptotically stable by Lyapunov. It is assumed that in the both cases the shock satisfies the well-known uniform Lopatinski condition [13]. Figure 1. Copyright © 2013 SciRes. OJAppS D. TKACHEV, A. BLOKHIN 80 The present work is a continuation of [10]. We con- sider the more general case when the Lopatinski condi- tion on the shock wave is satisfied only in a weak sense [13], i.e. the uniform Lopatinski condition can be vio- lated. In particular, this makes finding the classical solu- tion of this problem more difficult. The key point of the work is the analysis of an explicit form of the solution and we extensively use the technique developed in [10]. 2. Statement of the Original and the Auxiliary Problems and Formulation of Main Results The linear problem of finding a supersonic stationary gas flow onto a planar infinite wedge can be stated as follows [14]. We seek for a solution of the system of acoustic equations 0, tx y AUBUC U (1) in the domain ,0, tantxy x that satisfies the fol- lowing boundary conditions at the shock wave () and on the wedge tan :yx 13 342 3 0, 0,, tan ; y ty uduuuuF FF u (2) 21 tn ,auu (3) and at it also satisfies the initial data 00 0,,( ,),0,.UxyUxyFyFy (4) Here 1234 1234 are smooth perturbations of the components of the velocity, the pressure and the entropy respectively; (,,) (, , ,): , , ,Utxy uuuuuuuu (, ) x Fty is a small displacement of the discontinuity front, and 0 ,00 0,Ft F (5) that means that we consider the case of a shock wave attached to the wedge's vertex. It is also assumed that the components of the vector of initial data are compactly supported, i.e. 2 0{( ,)|,0},1,2,3,4. i suppuRx yx yi The matrices read as follows: 2 2 22 010 00 ( ,,1,1),, 1010 0001 M M AdiagMM B 0 ;A 0000 0010 ,tan 0100 0000 CCC 0 0 , u MM c1 is the Mach number behind of the shock wave (00 0 ,tanuv u are the components of the velocity vector of the stationary solution, 0 is the downstream sound speed), and the physical constants c ,,d were described in detail in [14]. These constants depend on the components of the piecewise constant so- lution corresponding to the step shock as well as on the state equation of gas (,)spp ( is the density of the gas and s is the entropy) and the equation of the Hugoniot adiabat. If the solution of problem (1)-(4) is continuous up to the boundary 0, tanxyx 0,0:tx y , then, in view of (5), it follows from the boundary conditions (2), (3) that the following compatibility condition should be fulfilled at the edge 2 3 (tan)( ,0,0)0,0,dut t i.e. if 2 1tan 0Dd , then (, 0, 0)0,0.Ut t (6) Remark 1. We have formulated the initial boundary-value prob- lems (1)-(5) for the case when the gas flow in a neighbor- hood of the wedge with shock wave directed along the - axis is fixed as the main solution (see Figure 2). Oy In this paper we consider the case of a weak shock wave, i.e. the case when 22 00 02 0 1. uv M Mcos c (7) Further we will also assume that the state equation of gas is such that 22 1 0,, (1)0,ddMM M (8) The coefficient 10D and, moreover, 1 (for example, this is true for the polytropic gas [10]; some examples of state equations satisfying (8) are considered in [15]). 0D Figure 2. Copyright © 2013 SciRes. OJAppS D. TKACHEV, A. BLOKHIN 81 Assume that the solution of problems (1)-(5) is not just continuous but also has second derivatives which are continuous up to the boundary of the domain. Taking mixed derivatives we can reduce problem (1)-(5) with condition (6) to the following initial boundary-value problem for the component (the pressure). In the domain 3 u 0, 0,tantxyx we seek for a smooth solution of the wave equation 2 222 12 3 0,ML Lu y (9) that satisfies the following boundary conditions at the shock wave () and on the wedge ( 0xtanyx ): 22 12 123 20;{}mLnLL Lu M (10) 30;cossin u yx (11) 3,0,0 0,ut (12) and it also satisfies the initial data for : 0t 300301 (, ),()(, )|| ttt uuxyu ux y 0 (13) (note that the derivative 3 can be found from the third equation of system (1)). ()| tt u In equations (9), (10) we have used the following no- tations: 2 1112 1,tan,M LllL l ty x 1 , 2 22 1, ,M Mnm d . The converse also holds, i.e. for each solution 3 of problems (9)-(13) we can uniquely find a corresponding smooth solution u (, ,),Utxy (, ) F ty of problem (1)-(6). This fact can be proved by the same way as for the case of a half-plane (see [14]). 0x Unlike [10], we assume that problem (9)—(13) satis- fies the Lopatinski condition [13] at the boundary 0x only in a weak sense, i.e. the uniform Lopatinski condi- tion (see [14]) or equivalently the condi- tion 0, 0mn 2 2 0, . M d (14) can be violated. In Figure 3 we shade the domain where the Lopatinski condition is satisfied (in a weak sense). This domain is described by the two systems of ine- qualities 2 2 2 2 1, 1,, and . 0 d M dd M M M d (15) The strait line 0 1 dM corresponds to the gas dynamic case, and and 0 are the densities ahead and behind of the shock front respectively. Remark 2. For the state equations from [15] the point (,)d lies in the second quadrant and 2 2. M d We point out that the uniform Lopatinski condition holds at the boundary tanyx . Now we make a convenient change of coordinates by setting ,taxxyyxn. (16) We shall drop the primes in what follows. Then problem (9)-(13) takes the following form: 22 2 tan 0,, ,0; {( )() ()} Mtx xy utxy y 2 (17) 2 2 tan tan 1tan0, 0; {( )[( ) ()]()} d tytxty ux xyy M (18) sin cos0,0;()uy yx (19) ,0,0 0;ut (20) 00 01 (, ),(, )|| ttt uuxyu uxy (21) Figure 3. Copyright © 2013 SciRes. OJAppS D. TKACHEV, A. BLOKHIN 82 (we also drop the subs the unknown function roblems (1)-(5) and (17)-(21) are equivalent, it is he following property ch cript 3 by 3 u). Since p sufficient to state our main results for the solution (, ,)utxy of problems (17)-(21). ition, we also assume that tIn add aracterizing the behavior of the solution as ,tx holds: there exist parameters 0 s and 0 p such function 00 (, ,) st px ee utxy isoundeas ,tx for each fixe that the bd d 0y, i.e. 00 (, ,), ,,0is fix () st px utxy Oe tx y ed (22) Let us introduce the notations . 2 01,M 0tan B 0 (1B), 2 1tanDd , 22tanDM ,0 0 1 1 B LB and assume that the problem's parameters are linked by the relation 21 21 || . DD L DD (23) The following results hold. ata for problem (17)-(21) are compactly su Theorem 1. If the initial d pported (i.e. 2 01 ,) s uppu uR and inequality (23) holds, then the cla the problem exists, is unique and can be defined by formula ssical solution of 0 00 () , 11 0 22 0 2 (,0) 22 0 2 22 8 (, ,) (, )*(()()tan),/ 2 8 ,( *(()tan), 8 / 2 / , () () ( ) ( Bxy xy BB t xy t t utxy B yx M M gtEty Bx xyBdlt M M Etyxxy d M 0 1 2 22 ,2(,) (()( )tan),/ 2 tan ,; 2,2 ) ( )() OQM P xydd u M Ety x , x yddxyx y x (24) the first integral is over the line , and the next two integrals are over the ab 0 yBx scissa axis y0 ; in the last two integrals over the qua we have the following coordinates: t 0), drangle he point 0POQM (,Qx y the point 0() , 22 () Bx y xy P , and the point 0(, ) M xy olution of t 1(, )uxy , ; the function is the fundamental she operatorn (14); and are the initial dre the es (, ,)Etxy of equatio ata (whe 0(, )uxy coordinat x y are expressed throuiables gh the var '' '' , x y); the functions (, ), g tx (, ),x lt (, ) f tx are known, in particular, (, ) f tx is determinllows: ed as fo (, )Ntx 1 , 0 (,)(,), px stn n f txLHps (2) where 5 (, )Ntx is a certain integer number, (, ) 0Ntx . Rem We do not write down here formulae defining th functions ark 3. e , (, ) f tx (, ) g tx , (, )ltx and (,) n H ps because of their awkwardness. Theorem 2. For problem (17)-(21) the trace ,)* ) 0 0 ( ,0) 0 2 00 2 0 2 (, )* *(()tan),/ 2 ,8tan (()( )tan), 2 ,2(,) (()( )tan),/ 2 ( )( ) ( ) ( xy t t OQM P OQM P ft y M Etyx xydu u M Ety x xydd u t M Ety x '' '' 0 2 (,)(, ,)|| x xy ftxutxy at the shock wave is a superposition of a fier of cylindrical and plane waves. Namely, repr nite numb esentation '' '' 0, (,0,)(,).| n (, ) 1 p xs txy n uLty Hps Nty takes place. Theorem 3. If y K (Kis a compact subset of the real half-axis), then there exists *(,)tK ( is the support of the initial data 0 u and chat (, )0fty 1 u) suh t when *(,),ttK y K . 3. Conclusions on the linearized level that the solution w when one shockont the Li condi e, i.e., the uniform Lopat- ins violated. ported initial data any So, 1) we prove ith a weak shock is asymptotically stable (by Lyapunov) in the case th fropatinsk- tions is satisfied in a weak sens ski condition i 2) Moreover, for compactly sup Copyright © 2013 SciRes. OJAppS D. TKACHEV, A. BLOKHIN Copyright © 2013 SciRes. OJAppS 83 solution of the linearized initial boundarye problem becomes stationary for a finite time. sta- ble whereas the weak shock wave solution is stable (when the Lopatinski condition holds at least a weak sense). 4. an th iaucation No. 14.B37.2.0355) also grateful to A.V. Yegitov for the ing the manuscript. ctures on Basic Gas Dynamics,” es of Flow Roun valu Our result justifies the classical Courant-Friedrichs' hypothesis that the strong shock wave solution is un in Acknowledgments This research was supported by the Russian Foundation for Basic Research (grts No. 10-01-302-a and 11-08- 00286-a) and e Russn Ministry of Ed (grant 1. The authors are assistance in prepar REFERENCES [1] R. Courant and K. O. Friedrichs, “Supersonic Flow and Shock Waves,” Intersc. P ubl., New York, 1948. [2] L. V. Ovsyannikov, “Le Institute of Computer Studies, Moscow, Izhevsk, 2003. [3] G. G. Chernyi, “Gas Dynamics,” Nauka, Moscow, 1988. [4] A. I. Rylov, “On the Possible Modd Ta- pered Bodies of Finite Thickness at Arbitrary Supersonic Velocities of the Approach Stream,” Journal of Applied Mathematic and Mechanics, Vol. 55, No. 1,1991, pp. 78-85.doi:10.1016/0021-8928(91)90065-3 [5] A. A. Nikolski, “Plane Vorticity Gas Flows,” Teor. Issle d. Mekhk. Zhidkosti i Gaza. Tr. 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