A singular limit problem for the Kudryashov-Sinelshchikov equation
aa r X i v : . [ m a t h . A P ] N ov A SINGULAR LIMIT PROBLEM FOR THEKUDRYASHOV-SINELSHCHIKOV EQUATION
GIUSEPPE MARIA COCLITE AND LORENZO DI RUVO
Abstract.
We consider the Kudryashov-Sinelshchikov equation, which contains nonlinear dis-persive effects. We prove that as the diffusion parameter tends to zero, the solutions of thedispersive equation converge to the entropy ones of the Burgers equation. The proof relies onderiving suitable a priori estimates together with an application of the compensated compactnessmethod in the L p setting. Introduction
A mixture of liquid and gas bubbles of the same size may be considered as an example ofa classic nonlinear medium. The analysis of propagation of the pressure waves in a liquidwith gas bubbles is an important problem. Indeed, there are solitary and periodic wavesin such mixtures and they can be described by nonlinear partial differential equations likethe Burgers, Korteweg-de Vries, and the Burgers-Korteweg-de Vries ones.Recently, Kudryashov and Sinelshchikov [7] obtained a more general nonlinear partialdifferential equation to describe the pressure waves in a liquid and gas bubbles mixturetaking into consideration the viscosity of liquid and the heat transfer. They introducedthe equation(1.1) ∂ t u + Au∂ x u + β∂ xxx u − Bβ∂ x (cid:0) u∂ xx u (cid:1) − Cβ∂ x u∂ xx u − ε∂ xx u − Dβ∂ x ( u∂ x u ) = 0 , where u is a density and models heat transfer and viscosity, while A, β, B, C, ε, D arereal parameters. If B = C = D = 0, (1.1) reads(1.2) ∂ t u + Au∂ x u + β∂ xxx u − ε∂ xx u = 0 , which is known as Korteweg-de Vries-Burgers equation [15]. If also ε = 0, we obtain theKorteweg-de Vries equation [6].Several results have been obtained in the case A = 1 , β = 1 , B = 1 , ε = 0 , D = 0 , in which (1.1) reads(1.3) ∂ t u + u∂ x u + ∂ xxx u − ∂ x (cid:0) u∂ xx u (cid:1) − C∂ x u∂ xx u = 0 . In [13], the author found four families of solitary wave solutions of (1.3) when C = − C = −
4. In [8], the authors discussed the existence of different kinds of traveling wavesolutions by using the approach of dynamical systems, according to different phase orbitsof the traveling system of (1.3); twenty-six kinds of exact traveling wave solutions areobtained under the parameter chioces C = − , − , ,
2. In [4], the authors discussed thebifurcations of phase portraits and investigated exact traveling wave solutions of (1.3) in
Date : September 2, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Singular limit, compensated compactness, Kudryashov-Sinelshchikov equation, entropycondition.The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). the cases C = − , ,
2. In [5], the authors investigated periodic loop solutions of (1.3),and discussed the limit forms of these solutions focusing on the case C = 2. In [12], theauthor studied (1.3) under the transformation α = 2 + C , in the cases α < α = 0,and α > C < − C = −
2, and
C > − α > C > meandering solutions was obtained. In [1], theauthors studied (1.3) by using the integral bifurcation method (see [14, 16]). They foundsome new traveling wave solutions of (1.3), which extends the results in [4, 5, 8, 12, 13].In this paper, we are interested to energy preserving waves, therefore we analyze (1.1)in the case(1.4) (
B, C ) = (cid:18) A , − A (cid:19) , D = 0 . We study the Cauchy problem(1.5) ∂ t u + Au∂ x u + β∂ xxx u − Bβ∂ x (cid:0) u∂ xx u (cid:1) − Cβ∂ x u∂ xx u − ε∂ xx u = 0 , t > , x ∈ R ,u (0 , x ) = u ( x ) , x ∈ R . We study the no high frequency limit, namely we send β, ε → ( ∂ t u + Au∂ x u = 0 , t > , x ∈ R ,u (0 , x ) = u ( x ) , x ∈ R . On the initial datum, we assume that(1.7) u ∈ L ( R ) ∩ L ( R ) . We study the dispersion-diffusion limit for (1.5). Therefore, we consider the followingthird order approximation(1.8) ∂ t u ε,β + Au ε,β ∂ x u ε,β + β∂ xxx u ε,β − Bβ∂ x (cid:0) u ε,β ∂ xx u ε,β (cid:1) − Cβ∂ x u ε,β ∂ xx u ε,β − ε∂ xx u ε,β = 0 , t > , x ∈ R ,u ε,β (0 , x ) = u ε,β, ( x ) , x ∈ R , where u ε,β, is a C ∞ approximation of u such that u ε, β, → u in L ploc ( R ), 1 ≤ p <
4, as ε, β → k u ε,β, k L ( R ) + k u ε,β, k L ( R ) + ( β + ε ) k ∂ x u ε,β, k L ( R ) ≤ C , ε, β > , (1.9)and C is a constant independent on ε and β .The main result of this paper is the following theorem. Theorem 1.1.
Assume that (1.7) and (1.9) hold. If (1.10) β = O ( ε ) , then, there exist two sequences { ε n } n ∈ N , { β n } n ∈ N , with ε n , β n → , and a limit function u ∈ L ∞ ( R + ; L ( R ) ∩ L ( R )) , such that u ε n ,β n → u strongly in L ploc ( R + × R ) , for each ≤ p < , (1.11) u is the unique entropy solution of (1.6) . (1.12) UDRYASHOV-SINELSHCHIKOV EQUATION 3
The paper is organized in four sections. In Section 2, we prove some a priori estimates,while in Section 3 we prove Theorem 1.1. In Appendix, we prove that Theorem 1.1 holdsalso in the case A = ( C + α ) n , where α is a suitable real number.2. A priori Estimates
This section is devoted to some a priori estimates on u ε,β . We denote with C theconstants which depend only on the initial data, and with C ( T ) the constants whichdepend also on T . Lemma 2.1.
Assume (1.4) . For each t > , k u ε,β ( t, · ) k L ( R ) + β k ∂ x u ε,β ( t, · ) k L ( R ) + 2 ε Z t k ∂ x u ε,β ( s, · ) k L ( R ) ds + 2 βε Z t (cid:13)(cid:13) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ C . (2.1) In particular, we have (2.2) k u ε,β ( t, · ) k L ∞ ( R ) ≤ C β − . Proof.
Multiplying (1.8) by u ε,β − β∂ xx u ε,β , we have( u ε,β − β∂ xx u ε,β ) ∂ t u ε,β + A ( u ε,β − β∂ xx u ε,β ) u ε,β ∂ x u ε,β + ( u ε,β − β∂ xx u ε,β ) β∂ xxx u ε,β − Bβ ( u ε,β − β∂ xx u ε,β ) ∂ x (cid:0) u ε,β ∂ xx u ε,β (cid:1) − Cβ ( u ε,β − β∂ xx u ε,β ) ∂ x u ε,β ∂ xx u ε,β − ε ( u ε,β − β∂ xx u ε,β ) ∂ xx u ε,β = 0 . (2.3)Since Z R (cid:0) u ε,β − β∂ xx u ε,β (cid:1) ∂ t u ε,β dx = 12 ddt (cid:16) k u ε,β ( t, · ) k L ( R ) + β k ∂ x u ε,β ( t, · ) k L ( R ) (cid:17) ,A Z R ( u ε,β − β∂ xx u ε,β ) u ε,β ∂ x u ε,β dx = − Aβ Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx,β Z R ( u ε,β − β∂ xx u ε,β ) β∂ xxx u ε,β dx = − β Z R ∂ x u ε,β ∂ xx u ε,β = 0 , − Bβ Z R ( u ε,β − β∂ xx u ε,β ) ∂ x (cid:0) u ε,β ∂ xx u ε,β (cid:1) dx = Bβ Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx + Bβ Z R ∂ xx u ε,β ∂ x (cid:0) u ε,β ∂ xx u ε,β (cid:1) dx, − Cβ Z R ( u ε,β − β∂ xx u ε,β ) ∂ x u ε,β ∂ xx u ε,β dx = − Cβ Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx + Cβ Z R ∂ x u ε,β ( ∂ xx u ε,β ) dx, − ε Z R ( u ε,β − β∂ xx u ε,β ) ∂ xx u ε,β dx = ε k ∂ x u ε,β ( t, · ) k L ( R ) + βε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) , G. M. COCLITE AND L. DI RUVO integrating (2.3) on R , we get ddt (cid:16) k u ε,β ( t, · ) k L ( R ) + β k ∂ x u ε,β ( t, · ) k L ( R ) (cid:17) + 2 ε k ∂ x u ε,β ( t, · ) k L ( R ) + 2 βε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) − β ( A − B + C ) Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx + 2 Bβ Z R ∂ xx u ε,β ∂ x (cid:0) u ε,β ∂ xx u ε,β (cid:1) dx + 2 Cβ Z R ∂ x u ε,β ( ∂ xx u ε,β ) dx = 0 . (2.4)Observe that2 Bβ Z R ∂ xx u ε,β ∂ x (cid:0) u ε,β ∂ xx u ε,β (cid:1) dx = − Bβ Z R u ε,β ∂ xxx u ε,β ∂ xx u ε,β dx = − Bβ Z R u ε,β ∂ x ( ∂ xx u ε,β ) dx = Bβ Z R ∂ x u ε,β ( ∂ xx u ε,β ) dx. Thus, from (2.4), ddt (cid:16) k u ε,β ( t, · ) k L ( R ) + β k ∂ x u ε,β ( t, · ) k L ( R ) (cid:17) + 2 ε k ∂ x u ε,β ( t, · ) k L ( R ) + 2 βε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) − β ( A − B + C ) Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx + β ( B + 2 C ) Z R ∂ x u ε,β ( ∂ xx u ε,β ) dx = 0 . Thanks to (1.4), we have(2.5) ( A − B + C = 0 ,B + 2 C = 0 . Therefore, (2.1) follows from (1.4), (1.9) and an integration on (0 , t ).Finally, we prove (2.2). Due to (2.1) and the H¨older inequality, u ε,β ( t, x ) =2 Z x −∞ u ε,β ∂ x u ε,β dx ≤ Z R | u ε,β || ∂ x u ε,β | dx ≤ k u ε,β ( t, · ) k L ( R ) k ∂ x u ε,β ( t, · ) k L ( R ) ≤ C β − . Therefore, | u ε,β ( t, x ) | ≤ C β − , which gives (2.2). (cid:3) Following [2, Lemma 2 . . Lemma 2.2.
Assume that (1.4) and (1.10) hold. Then, for each t > , i ) the family { u ε,β } ε,β is bounded in L ∞ ( R + ; L ( R )) ; ii ) the family { ε∂ x u ε,β } ε,β is bounded in L ∞ ( R + ; L ( R )) ; iii ) the families {√ εu ε,β ∂ x u ε,β } ε,β , { ε √ ε∂ xx u ε,β } ε,β are bounded in L ( R + × R ) .Moreover, β Z t (cid:13)(cid:13) ∂ x u ε,β ( s, · ) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ C ε , t > , (2.6) β Z t (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) ds ≤ C ε , t > , (2.7) UDRYASHOV-SINELSHCHIKOV EQUATION 5 β Z t (cid:13)(cid:13) u ε,β ( s, · ) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ C ε , t > , (2.8) β Z t (cid:13)(cid:13) u ε,β ( s, · ) ∂ x u ε,β ( s, · ) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ C ε, t > . (2.9) Proof.
Let K be a positive constant which will be specified later. Multiplying (1.8) by Ku ε,β − ε ∂ xx u ε,β , we have (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) ∂ t u ε,β + A (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) u ε,β ∂ x u ε,β + β (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) ∂ xxx u ε,β − Bβ (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) ∂ x (cid:0) u∂ xx u (cid:1) − Cβ (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) ∂ x u ε,β ∂ xx u ε,β − ε (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) ∂ xx u ε,β = 0 . (2.10)Observe that Z R (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) ∂ t u ε,β dx = ddt (cid:18) K k u ε,β ( t, · ) k L ( R ) + ε k ∂ x u ε,β ( t, · ) k L ( R ) (cid:19) ,A Z R (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) u ε,β ∂ x u ε,β = − Aε Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx,β Z R (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) ∂ xxx u ε,β dx = − Kβ Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx, − Bβ Z R (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) ∂ x (cid:0) u∂ xx u (cid:1) dx =3 BKβ Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx + ε βB Z R u ε,β ∂ xx u ε,β ∂ xxx u ε,β dx =3 BKβ Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx + ε βB Z R u ε,β ( ∂ xx u ε,β ) dx, − Cβ Z R (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) ∂ x u ε,β ∂ xx u ε,β dx = − CKβ Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx + ε βC Z R u ε,β ( ∂ xx u ε,β ) dx, − ε Z R (cid:0) Ku ε,β − ε ∂ xx u ε,β (cid:1) ∂ xx u ε,β dx =3 Kε k u ε,β ( t, · ) ∂ x u ε,β ( t, · ) k L ( R ) + ε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) . G. M. COCLITE AND L. DI RUVO
Therefore, integrating (2.10) over R , from (1.4), we get ddt (cid:18) K k u ε,β ( t, · ) k L ( R ) + ε k ∂ x u ε,β ( t, · ) k L ( R ) (cid:19) + 3 Kε k u ε,β ( t, · ) ∂ x u ε,β ( t, · ) k L ( R ) + ε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) = − Aε Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx + 3 Kβ Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx + 7 A Kβ Z R u ε,β ∂ x u ε,β ∂ xx u ε,β dx ≤ ε | A | Z R | u ε,β ∂ x u ε,β || ∂ xx u ε,β | dx + 3 Kβ Z R u ε,β | ∂ x u ε,β || ∂ xx u ε,β | dx + 73 Kβ Z R (cid:12)(cid:12) Au ε,β ∂ x u ε,β (cid:12)(cid:12) | ∂ xx u ε,β | dx. (2.11)Due to the Young inequality, ε | A | Z R | u ε,β ∂ x u ε,β || ∂ xx u ε,β | dx = Z R (cid:12)(cid:12)(cid:12) ε √ Au ε,β ∂ x u ε,β (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε √ ∂ xx u ε,β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ εA k u ε,β ( t, · ) ∂ x u ε,β ( t, · ) k L ( R ) + ε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) . Hence, from (2.11), ddt (cid:18) K k u ε,β ( t, · ) k L ( R ) + ε k ∂ x u ε,β ( t, · ) k L ( R ) (cid:19) + ε (cid:18) K − A (cid:19) k u ε,β ( t, · ) ∂ x u ε,β ( t, · ) k L ( R ) + 5 ε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) ≤ Kβ Z R u ε,β | ∂ x u ε,β || ∂ xx u ε,β | dx + 73 Kβ Z R | Au ε,β || ∂ x u ε,β || ∂ xx u ε,β | dx. (2.12)Observe that, from (1.10),(2.13) β ≤ D ε , where D is a positive constant which will be specified later. It follows from (2.2), (2.13)and the Young inequality that3 Kβ Z R u ε,β | ∂ x u ε,β || ∂ xx u ε,β | dx ≤ Kβ k u ε,β ( t, · ) k L ∞ ( R ) Z R | ∂ x u ε,β || ∂ xx u ε,β | dx ≤ KC β Z R | ∂ x u ε,β || ∂ xx u ε,β | dx ≤ Z R (cid:12)(cid:12)(cid:12) √ C D Kε ∂ x u ε,β (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε √ ∂ xx u ε,β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ C D K ε k ∂ x u ε,β ( t, · ) k L ( R ) + ε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) , Kβ Z R | Au ε,β || ∂ x u ε,β || ∂ xx u ε,β | dx ≤ Kβ k u ε,β ( t, · ) k L ∞ ( R ) Z R | Au ε,β || ∂ x u ε,β || ∂ xx u ε,β | dx ≤ KC β Z R | Au ε,β || ∂ x u ε,β || ∂ xx u ε,β | dx ≤ Z R (cid:12)(cid:12)(cid:12) √ C AD Kε u ε,β ∂ x u ε,β (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε √ ∂ xx u ε,β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ C A D K ε k u ε,β ( t, · ) ∂ x u ε,β ( t, · ) k L ( R ) + ε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) . UDRYASHOV-SINELSHCHIKOV EQUATION 7
Therefore, we have ddt (cid:18) K k u ε,β ( t, · ) k L ( R ) + ε k ∂ x u ε,β ( t, · ) k L ( R ) (cid:19) + ε (cid:18) K − A (cid:19) k u ε,β ( t, · ) ∂ x u ε,β ( t, · ) k L ( R ) + 5 ε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) ≤ C D K ε k ∂ x u ε,β ( t, · ) k L ( R ) + ε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) + C A D K ε k u ε,β ( t, · ) ∂ x u ε,β ( t, · ) k L ( R ) , that is ddt (cid:18) K k u ε,β ( t, · ) k L ( R ) + ε k ∂ x u ε,β ( t, · ) k L ( R ) (cid:19) + ε (cid:18) K − A − C A D K (cid:19) k u ε,β ( t, · ) ∂ x u ε,β ( t, · ) k L ( R ) + ε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) ≤ C D K ε k ∂ x u ε,β ( t, · ) k L ( R ) . (2.14)We search a constant K such that(2.15) C A D K − K + 3 A < .K does exist if and only if(2.16) 3 − C A D > . Choosing(2.17) D = 1 √ C A , it follows from (2.15) and (2.17) that, there exist 0 < K < K , such that for every(2.18) K < K < K (2.15) holds. Hence, from (2.18), choosing K < K < K , we get ddt (cid:18) K k u ε,β ( t, · ) k L ( R ) + ε k ∂ x u ε,β ( t, · ) k L ( R ) (cid:19) + εK k u ε,β ( t, · ) ∂ x u ε,β ( t, · ) k L ( R ) + ε (cid:13)(cid:13) ∂ xx u ε,β ( t, · ) (cid:13)(cid:13) L ( R ) ≤ K ε k ∂ x u ε,β ( t, · ) k L ( R ) , (2.19)where K and K are two fixed positive constants. Integrating (2.19) on (0 , t ), from (1.9)and (2.1), we have K k u ε,β ( t, · ) k L ( R ) + ε k ∂ x u ε,β ( t, · ) k L ( R ) + εK Z t k u ε,β ( s, · ) ∂ x u ε,β ( s, · ) k L ( R ) ds + ε Z t (cid:13)(cid:13) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ C + K ε Z t k ∂ x u ε,β ( s, · ) k L ( R ) ds ≤ C (1 + K ) ≤ C . (2.20) G. M. COCLITE AND L. DI RUVO
Then, k u ε,β ( t, · ) k L ( R ) ≤ C ,ε k ∂ x u ε,β ( t, · ) k L ( R ) ≤ C ,ε Z t k u ε,β ( s, · ) ∂ x u ε,β ( s, · ) k L ( R ) ds ≤ C ,ε Z t (cid:13)(cid:13) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ C , (2.21)for every t >
0. Thanks to (2.1), (2.13), (2.21) and the H¨older inequality, β Z t (cid:13)(cid:13) ∂ x u ε,β ( s, · ) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds = βε Z t Z R ε | ∂ x u ε,β | ε | ∂ xx | dsdx ≤ βε (cid:18) ε Z t k ∂ x u ε,β ( s, · ) k L ( R ) ds (cid:19) (cid:18) ε Z t (cid:13)(cid:13) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds (cid:19) ≤ C βε ≤ C D ε , that is (2.6). Due to (2.13) and (2.21), β Z t (cid:13)(cid:13) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds = β ε ε Z t (cid:13)(cid:13) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ C D ε , which gives (2.7). It follows from (2.2), (2.13) and (2.21) that β Z t (cid:13)(cid:13) u ε,β ( s, · ) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ β k u ε,β k L ∞ ((0 , ∞ ) × R ) Z t (cid:13)(cid:13) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ β ε ε Z t (cid:13)(cid:13) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ C D ε ε ≤ C ε , that is (2.8). From (2.1), (2.2), (2.13), (2.21) and the H¨older inequality, β Z t (cid:13)(cid:13) u ε,β ( s, · ) ∂ x u ε,β ( s, · ) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ β k u ε,β k L ∞ ((0 , ∞ ) × R ) Z t Z R | ∂ x u ε,β || ∂ xx u ε,β | dsdx ≤ C β ε Z t Z R ε | ∂ x u ε,β | ε | ∂ xx u ε,β | dsdx ≤ C β ε (cid:18) ε Z t k ∂ x u ε,β ( s, · ) k L ( R ) ds (cid:19) (cid:18) ε Z t (cid:13)(cid:13) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds (cid:19) ≤ C D ε ε ≤ C ε, which gives (2.9). (cid:3) Proof of Theorem 1.1
In this section, we prove Theorem 1.1. The following technical lemma is needed [10].
Lemma 3.1.
Let Ω be a bounded open subset of R . Suppose that the sequence {L n } n ∈ N of distributions is bounded in W − , ∞ (Ω) . Suppose also that L n = L ,n + L ,n , UDRYASHOV-SINELSHCHIKOV EQUATION 9 where {L ,n } n ∈ N lies in a compact subset of H − loc (Ω) and {L ,n } n ∈ N lies in a boundedsubset of M loc (Ω) . Then {L n } n ∈ N lies in a compact subset of H − loc (Ω) . Moreover, we consider the following definition.
Definition 3.1.
A pair of functions ( η, q ) is called an entropy–entropy flux pair if η : R → R is a C function and q : R → R is defined by q ( u ) = Z u Aξη ′ ( ξ ) dξ. An entropy-entropy flux pair ( η, q ) is called convex/compactly supported if, in addition, η is convex/compactly supported. Following [9], we prove Theorem 1.1.
Proof of Theorem 1.1.
Let us consider a compactly supported entropy–entropy flux pair( η, q ). Multiplying (1.8) by η ′ ( u ε,β ), we have ∂ t η ( u ε,β ) + ∂ x q ( u ε,β ) = εη ′ ( u ε,β ) ∂ xx u ε,β − βη ′ ( u ε,β ) ∂ xxx u ε,β − Bβη ′ ( u ε,β ) ∂ x (cid:0) u ε,β ∂ xx u ε,β (cid:1) − Cβη ′ ( u ε,β ) ∂ x u ε,β ∂ xx u ε,β = I , ε, β + I , ε, β + I , ε, β + I , ε, β + I , ε, β + I , ε, β + I , ε, β , where I , ε, β = ∂ x ( εη ′ ( u ε,β ) ∂ x u ε,β ) ,I , ε, β = − εη ′′ ( u ε,β )( ∂ x u ε,β ) ,I , ε, β = ∂ x ( − βη ′ ( u ε,β ) ∂ xx u ε,β ) ,I , ε, β = βη ′′ ( u ε,β ) ∂ x u ε,β ∂ xx u ε,β ,I , ε, β = ∂ x (cid:0) − Bβη ′ ( u ε,β ) u ε,β ∂ xx u ε,β (cid:1) ,I , ε, β = Bβη ′′ ( u ε,β ) u ε,β ∂ x u ε,β ∂ xx u ε,β ,I , ε, β = − Cβη ′ ( u ε,β ) ∂ x u ε,β ∂ xx u ε,β . (3.1)We have I , ε, β → H − ((0 , T ) × R ) , T >
0, as ε → (cid:13)(cid:13) εη ′ ( u ε,β ) ∂ x u ε,β (cid:13)(cid:13) L ((0 ,T ) × R )) ≤ (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) ε Z T k ∂ x u ε,β ( s, · ) k L ( R ) ds ≤ (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) εC → . We claim that { I , ε, β } ε, β> is bounded in L ((0 , T ) × R ) , T > . Again by Lemma 2.1, (cid:13)(cid:13) εη ′′ ( u ε,β )( ∂ x u ε,β ) (cid:13)(cid:13) L ((0 ,T ) × R ) ≤ (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) ε Z T k ∂ x u ε,β ( s, · ) k L ( R ) ds ≤ (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) C . We have that I , ε, β → H − ((0 , T ) × R ) , T > , as ε → Thanks to Lemma 2.2, (cid:13)(cid:13) β η ′ ( u ε,β ) ∂ xx u ε,β (cid:13)(cid:13) L ((0 ,T ) × R )) ≤ (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) β Z T (cid:13)(cid:13) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) C ( T ) ε → . We show that I , ε, β → L ((0 , T ) × R ) , T > , as ε → (cid:13)(cid:13) βη ′′ ( u ε,β ) ∂ x u ε,β ∂ xx u ε,β (cid:13)(cid:13) L ((0 ,T ) × R ) ≤ (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) β Z T (cid:13)(cid:13) ∂ x u ε,β ( s, · ) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) C ε → . We claim that I , ε, β → H − ((0 , T ) × R ) , T > , as ε → (cid:13)(cid:13) Bβη ′ ( u ε,β ) u ε,β ∂ xx u ε,β (cid:13)(cid:13) L ((0 ,T ) × R ) ≤ B β (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) β Z T (cid:13)(cid:13) u ε,β ( s, · ) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ B (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) C ε → . We have that I , ε, β → L ((0 , T ) × R ) , T > , as ε → (cid:13)(cid:13) Bβη ′′ ( u ε,β ) u ε,β ∂ x u ε,β ∂ xx u ε,β (cid:13)(cid:13) L ((0 ,T ) × R ) ≤ | B | (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) β Z T (cid:13)(cid:13) u ε,β ( s, · ) ∂ x u ε,β ( s, · ) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ | B | (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) C ε → . We claim that I , ε, β → L ((0 , T ) × R ) , T > , as ε → (cid:13)(cid:13) Cβη ′ ( u ε,β ) ∂ x u ε,β ∂ xx u ε,β (cid:13)(cid:13) L ((0 ,T ) × R ) ≤ | C | (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) β Z T (cid:13)(cid:13) ∂ x u ε,β ( s, · ) ∂ xx u ε,β ( s, · ) (cid:13)(cid:13) L ( R ) ds ≤ | C | (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) C ε → . Therefore, (1.11) follows from Lemma 3.1 and the L p compensated compactness [11].We have to show that (1.12) holds. We begin by proving that u is a distributionalsolution of (1.6). Let φ ∈ C ∞ ( R ) be a test function with compact support. We have toprove that(3.2) Z ∞ Z R (cid:18) u∂ t φ + Au ∂ x φ (cid:19) dtdx + Z R u ( x ) φ (0 , x ) dx = 0 . UDRYASHOV-SINELSHCHIKOV EQUATION 11
We have that Z ∞ Z R u ε n ,β n ∂ t φ + Au ε n ,β n ∂ x φ ! dtdx + Z R u ,ε n ,β n ( x ) φ (0 , x ) dx + ε n Z ∞ Z R u ε n ,β n ∂ xx φdtdx + ε n Z ∞ u ,ε n ,β n ( x ) ∂ xx φ (0 , x ) dx + β n Z ∞ Z R u ε n ,β n ∂ xxx φdtdx + β n Z ∞ u ,ε n ,β n ( x ) ∂ xxx φ (0 , x ) dx = Bβ n Z ∞ Z R u ε n ,β n ∂ xx u ε n ,β n ∂ x φdtdx − Cβ n Z ∞ Z R ∂ x u ε n ,β n ∂ xx u ε n ,β n φdtdx. (3.3)Let us show that(3.4) Bβ n Z ∞ Z R u ε n ,β n ∂ xx u ε n ,β n ∂ x φdtdx → . Fix
T >
0. Due to (1.10), (2.2), Lemma 2.2 and the H¨older inequality, | B | β n (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ Z R u ε n ,β n ∂ xx u ε n ,β n ∂ x φdtdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ | B | β n Z ∞ Z R | u ε n ,β n || ∂ xx u ε n ,β n || ∂ x φ | dtdx ≤ | B | β n k u ε n ,β n k L ∞ (0 , ∞ ) × R ) Z ∞ Z R | ∂ xx u ε n ,β n || ∂ x φ | dtdx ≤ | B | C β n (cid:13)(cid:13) ∂ xx u ε n ,β n (cid:13)(cid:13) L (supp ( ∂ x φ )) k ∂ x φ k L (supp ( ∂ x φ )) ≤ | B | C ε n (cid:13)(cid:13) ∂ xx u ε n ,β n (cid:13)(cid:13) L ((0 ,T ) × R ) k ∂ x φ k L ((0 ,T ) × R ) ≤ | B | C ε n → , that is (3.4).We prove that(3.5) − Cβ n Z ∞ Z R ∂ x u ε n ,β n ∂ xx u ε n ,β n φdtdx → . Fix
T >
0. Thanks to Lemma 2.2, | C | β n (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ Z R ∂ x u ε n ,β n ∂ xx u ε n ,β n φdtdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ | C | β n Z ∞ Z R | ∂ x u ε n ,β n ∂ xx u ε n ,β n || φ | dtdx ≤ | C | k φ k L ∞ (supp ( φ )) β n (cid:13)(cid:13) ∂ x u ε n ,β n ∂ xx u ε n ,β n (cid:13)(cid:13) L (supp ( φ )) ≤ | C | k φ k L ∞ ((0 ,T ) × R ) β n (cid:13)(cid:13) ∂ x u ε n ,β n ∂ xx u ε n ,β n (cid:13)(cid:13) L ((0 ,T ) × R ) ≤ | C | k φ k L ∞ ((0 ,T ) × R ) C ε → , which gives (3.5). Therefore, (3.2) follows from (1.9), (1.11), (3.3), (3.4) and (3.5).We conclude by proving that u is the unique entropy solution of (1.6). Fix T >
0. Let usconsider a compactly supported entropy–entropy flux pair ( η, q ), and φ ∈ C ∞ c ((0 , ∞ ) × R )a non–negative function. We have to prove that(3.6) Z ∞ Z R ( ∂ t η ( u ) + ∂ x q ( u )) φdtdx ≤ . We have Z ∞ Z R ( ∂ x η ( u ε n , β n ) + ∂ x q ( u ε n , β n )) φdtdx = ε n Z ∞ Z R ∂ x ( η ′ ( u ε n , β n ) ∂ x u ε n , β n ) φdtdx − ε n Z ∞ Z R η ′′ ( u ε n , β n )( ∂ x u ε n , β n ) φdtdx − β n Z ∞ Z R ∂ x ( η ′ ( u ε n , β n ) ∂ xx u ε n , β n ) φdtdx + β n Z ∞ Z R η ′′ ( u ε n , β n ) ∂ x u ε n , β n ∂ xx u ε n , β n φdtdx − Bβ n Z ∞ Z R ∂ x ( η ′ ( u ε n , β n ) u ε n , β n ∂ xx u ε n , β n ) φdtdx + Bβ n Z ∞ Z R η ′′ ( u ε n , β n ) u ε n , β n ∂ x u ε n , β n ∂ xx u ε n , β n φdtdx − Cβ n Z ∞ Z R η ′ ( u ε n , β n ) ∂ x u ε n , β n ∂ xx u ε n , β n φdtdx ≤ − ε n Z ∞ Z R η ′ ( u ε n , β n ) ∂ x u ε n , β n ∂ x φdtdx + β n Z ∞ Z R η ′ ( u ε n , β n ) ∂ xx u ε n , β n ∂ x φdtdx + β n Z ∞ Z R η ′′ ( u ε n , β n ) ∂ x u ε n , β n ∂ xx u ε n , β n φdtdx + Bβ n Z ∞ Z R η ′ ( u ε n , β n ) u ε n , β n ∂ xx u ε n , β n ∂ x φdtdx + Bβ n Z ∞ Z R η ′′ ( u ε n , β n ) u ε n , β n ∂ x u ε n , β n ∂ xx u ε n , β n φdtdx − Cβ n Z ∞ Z R η ′ ( u ε n , β n ) ∂ x u ε n , β n ∂ xx u ε n , β n φdtdx ≤ ε n Z ∞ Z R | η ′ ( u ε n , β n ) || ∂ x u ε n , β n || ∂ x φ | dtdx + β n Z ∞ Z R | η ′ ( u ε n , β n ) || ∂ xx u ε n , β n || ∂ x φ | dtdx + β n Z ∞ Z R | η ′′ ( u ε n , β n ) || ∂ x u ε n , β n ∂ xx u ε n , β n || φ | dtdx + | B | β n Z ∞ Z R | η ′ ( u ε n , β n ) || u ε n , β n ∂ xx u ε n , β n || ∂ x φ | dtdx + | B | β n Z ∞ Z R | η ′′ ( u ε n , β n ) || u ε n , β n ∂ x u ε n , β n ∂ xx u ε n , β n || φ | dtdx + | C | β n Z ∞ Z R | η ′ ( u ε n , β n ) || ∂ x u ε n , β n ∂ xx u ε n , β n || φ | dtdx. Hence, from (2.2), Z ∞ Z R ( ∂ x η ( u ε n , β n ) + ∂ x q ( u ε n , β n )) φdtdx UDRYASHOV-SINELSHCHIKOV EQUATION 13 ≤ ε n (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) k ∂ x u ε n , β n k L (supp ( ∂ x φ )) k ∂ x φ k L (supp ( ∂ x φ )) + β n (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) (cid:13)(cid:13) ∂ xx u ε n , β n (cid:13)(cid:13) L (supp ( ∂ x φ )) k ∂ x φ k L (supp ( ∂ x φ )) + β n (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) k φ k L ∞ ( R ) (cid:13)(cid:13) ∂ x u ε n , β n ∂ xx u ε n , β n (cid:13)(cid:13) L (supp ( ∂ x φ )) + | B | (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) β n k u ε n , β n k L ∞ ((0 , ∞ ) × R ) Z ∞ Z R | ∂ xx u ε n , β n || ∂ x φ | dtdx + | B | (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) β n k u ε n , β n k L ∞ ((0 , ∞ ) × R ) Z ∞ Z R | ∂ x u ε n , β n || ∂ xx u ε n , β n || ∂ x φ | dtdx + β n | C | (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) k φ k L ∞ ( R ) (cid:13)(cid:13) ∂ x u ε n , β n ∂ xx u ε n , β n (cid:13)(cid:13) L (supp ( ∂ x φ )) ≤ ε n (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) k ∂ x u ε n , β n k L ((0 ,T ) × R ) k ∂ x φ k L ((0 ,T ) × R ) + β n (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) (cid:13)(cid:13) ∂ xx u ε n , β n (cid:13)(cid:13) L ((0 ,T ) × R ) k ∂ x φ k L ((0 ,T ) × R ) + β n (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) k φ k L ∞ ( R + × R ) (cid:13)(cid:13) ∂ x u ε n , β n ∂ xx u ε n , β n (cid:13)(cid:13) L ((0 ,T ) × R ) + β n | C | (cid:13)(cid:13) η ′ (cid:13)(cid:13) L ∞ ( R ) β n k φ k L ∞ ( R + × R ) (cid:13)(cid:13) ∂ x u ε n , β n ∂ xx u ε n , β n (cid:13)(cid:13) L ((0 ,T ) × R ) + C | B | (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) β n Z ∞ Z R | ∂ xx u ε n , β n || ∂ x φ | dtdx + C | B | (cid:13)(cid:13) η ′′ (cid:13)(cid:13) L ∞ ( R ) β n Z ∞ Z R | ∂ x u ε n , β n || ∂ xx u ε n , β n || ∂ x φ | dtdx, that is Z ∞ Z R ( ∂ x η ( u ε n , β n ) + ∂ x q ( u ε n , β n )) φdtdx ≤ C ε n k ∂ x u ε n , β n k L ((0 ,T ) × R ) + C β n (cid:13)(cid:13) ∂ xx u ε n , β n (cid:13)(cid:13) L ((0 ,T ) × R ) + C β n (cid:13)(cid:13) ∂ x u ε n , β n ∂ xx u ε n , β n (cid:13)(cid:13) L ((0 ,T ) × R ) + C β n Z ∞ Z R | ∂ xx u ε n , β n || ∂ x φ | dtdx + C β n Z ∞ Z R | ∂ x u ε n , β n || ∂ xx u ε n , β n || ∂ x φ | dtdx, (3.7)where C is a suitable positive constant.Let us show that(3.8) β n Z ∞ Z R | ∂ xx u ε n , β n || ∂ x φ | dtdx → . Due (1.10), Lemma (2.2) and the H¨older inequality, β n Z ∞ Z R | ∂ xx u ε n , β n || ∂ x φ | dtdx ≤ C ε n (cid:13)(cid:13) ∂ xx u ε n , β n (cid:13)(cid:13) L (supp ( ∂ x φ )) k ∂ x φ k L (supp ( ∂ x φ )) ≤ C ε n (cid:13)(cid:13) ∂ xx u ε n , β n (cid:13)(cid:13) L ((0 ,T ) × R ) k ∂ x φ k L ((0 ,T ) × R ) ≤ C ε n k ∂ x φ k L ((0 ,T ) × R ) → , that is (3.8).We claim that(3.9) β n Z ∞ Z R | ∂ x u ε n , β n || ∂ xx u ε n , β n || ∂ x φ | dtdx → . Thanks to Lemmas 2.1, 2.2 and the H¨older inequality, β n Z ∞ Z R | ∂ x u ε n , β n || ∂ xx u ε n , β n || ∂ x φ | dtdx ≤ C ε n k φ k L ∞ ( R + × R ) (cid:13)(cid:13) ∂ x u ε n , β n ∂ xx u ε n , β n (cid:13)(cid:13) L (supp ( φ )) ≤ C k φ k L ∞ ( R + × R ) ε n Z T Z R ε n | ∂ x u ε n , β n | ε n | ∂ xx u ε n , β n | dtdx ≤ C k φ k L ∞ ( R + × R ) ε n (cid:18) ε n Z T k ∂ x u ε n , β n ( t, · ) k L ( R ) dt (cid:19) · (cid:18) ε n Z T (cid:13)(cid:13) ∂ xx u ε n , β n ( t, · ) (cid:13)(cid:13) L ( R ) dt (cid:19) ≤ C k φ k L ∞ ( R + × R ) ε → , which gives (3.8).Finally, (3.6) follows from (1.10), (1.11), (3.7), (3.8), (3.9) and Lemmas 2.1 and 2.2. (cid:3) Appendix A. On the case A = ( C + α ) n Theorem 1.1 holds also in the cases A = C and A = C n , with C = 0 and n ∈ N .Indeed, from (2.5), if A = C , we get C = −
3, while if A = C n , we obtain C = − n − .If A = C n +1 , from (2.5), we get C n + 3 = 0 , which does not have solutions in R .In this section, we prove that Theorem 1.1 holds also in the case A = ( C + α ) n , where α is a suitable real number. We only need to prove the following result Lemma A.1.
Assume that (A.1) A = ( C + α ) n . If (A.2) α ≤ n − (cid:18) n (cid:19) n n − + (cid:18) n (cid:19) n − , then (2.1) holds.Proof. We begin by observing that, by (2.5), we have( C + α ) n + 3 C = 0 , that is(A.3) ( C + α ) n + 3 ( C + α ) − α = 0 . Let us consider the following function(A.4) g ( X ) = X n + 3 X − α. We observe that(A.5) lim X →−∞ g ( X ) = ∞ , lim X →∞ g ( X ) = ∞ . UDRYASHOV-SINELSHCHIKOV EQUATION 15
Since g ′ ( X ) = 2 nX n − + 3, we have that(A.6) g is increasing in − (cid:18) n (cid:19) n − , ∞ ! . From (A.2),(A.7) g ( X ) ≤ , X = − (cid:18) n (cid:19) n − . Then, it follows from (A.5), (A.6) and (A.7) that the function g has only two zeros X < < X . Hence, from (A.1), A = X n , or A = X n .Therefore, arguing as in Lemma 2.1, we have (2.1). (cid:3) References [1]
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Department of Mathematics, University of Bari, via E. Orabona 4, 70125 Bari, Italy
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