Topological resolution of Riemann problems for pairs of conservation laws

Authors:
Arthur V. Azevedo, Cesar S. Eschenazi, Dan Marchesin and Carlos F. B. Palmeira

Journal:
Quart. Appl. Math. **68** (2010), 375-393

MSC (2000):
Primary 35L65, 58J45, 35L60, 57R99

Published electronically:
February 19, 2010

MathSciNet review:
2663005

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Abstract: The structure of Riemann solutions to certain systems of conservation laws can be so complicated that the classical constructions are unable to establish their global existence and stability. For systems of two conservation laws, classically the local solution is found by intersecting two wave curves specified by the Riemann data. The intersection point represents the intermediate constant state that typically appears in such solutions. In this paper, we construct the wave curves in a three-dimensional manifold which is globally foliated by shock curves, and where rarefaction and composite curves are naturally defined. The main innovation in this paper is the construction of a differentiable two-dimensional manifold of intermediate states; the local classical construction is replaced by finding the intersection of a wave curve with the intermediate manifold; a transversality argument guarantees the stability of the Riemann solution. The new construction has the potential of establishing structural stability properties globally, i.e., for all initial data. This is its main advantage over the classical construction, which is intrinsically local. The construction presented in this work is demonstrated for a family of quadratic polynomial flux functions and their perturbations.

**1.**V. I. Arnold,*Mathematical methods of classical mechanics*, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60. MR**0690288****2.**J. Basto-Gonçalves and H. Reis,*The geometry of 2×2 systems of conservation laws*, Acta Appl. Math.**88**(2005), no. 3, 269–329. MR**2192058**, 10.1007/s10440-005-9002-5**3.**C. S. Eschenazi,*Rarefaction fields in systems of two and three conservation laws*, Doctoral thesis, Dep. de Matemática, PUC-Rio, 1992 (in Portuguese).**4.**Cesar S. Eschenazi and Carlos Frederico B. Palmeira,*The structure of composite rarefaction-shock foliations for quadratic systems of conservation laws*, Mat. Contemp.**22**(2002), 113–140. Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). MR**1965790****5.**F. Furtado,*Structural stability of nonlinear waves for conservation laws*, Ph.D. thesis, New York Univ., 1989.**6.**I. M. Gel′fand,*Some problems in the theory of quasilinear equations*, Amer. Math. Soc. Transl. (2)**29**(1963), 295–381. MR**0153960****7.**J. Sotomayor and C. Gutierrez,*Structurally stable configurations of lines of principal curvature*, Bifurcation, ergodic theory and applications (Dijon, 1981) Astérisque, vol. 98, Soc. Math. France, Paris, 1982, pp. 195–215. MR**724448****8.**Eli L. Isaacson, Dan Marchesin, C. Frederico Palmeira, and Bradley J. Plohr,*A global formalism for nonlinear waves in conservation laws*, Comm. Math. Phys.**146**(1992), no. 3, 505–552. MR**1167301****9.**Tai Ping Liu,*The Riemann problem for general 2×2 conservation laws*, Trans. Amer. Math. Soc.**199**(1974), 89–112. MR**0367472**, 10.1090/S0002-9947-1974-0367472-1**10.**Dan Marchesin and C. Frederico B. Palmeira,*Topology of elementary waves for mixed-type systems of conservation laws*, J. Dynam. Differential Equations**6**(1994), no. 3, 427–446. MR**1298810**, 10.1007/BF02218857**11.**Stephen Schecter, Dan Marchesin, and Bradley J. Plohr,*Structurally stable Riemann solutions*, J. Differential Equations**126**(1996), no. 2, 303–354. MR**1383980**, 10.1006/jdeq.1996.0053**12.**Stephen Schecter, Bradley J. Plohr, and Dan Marchesin,*Classification of codimension-one Riemann solutions*, J. Dynam. Differential Equations**13**(2001), no. 3, 523–588. MR**1845094**, 10.1023/A:1016634307145**13.**Ralph Menikoff and Bradley J. Plohr,*The Riemann problem for fluid flow of real materials*, Rev. Modern Phys.**61**(1989), no. 1, 75–130. MR**977944**, 10.1103/RevModPhys.61.75**14.**O. A. Oleĭnik,*On the uniqueness of the generalized solution of the Cauchy problem for a non-linear system of equations occurring in mechanics*, Uspehi Mat. Nauk (N.S.)**12**(1957), no. 6(78), 169–176 (Russian). MR**0094543****15.**C. F. B. Palmeira,*Line fields defined by eigenspaces of derivatives of maps from the plane to itself*, Proceedings of the Sixth International Colloquium on Differential Geometry (Santiago de Compostela, 1988) Cursos Congr. Univ. Santiago de Compostela, vol. 61, Univ. Santiago de Compostela, Santiago de Compostela, 1989, pp. 177–205. MR**1040846****16.**David G. Schaeffer and Michael Shearer,*The classification of 2×2 systems of nonstrictly hyperbolic conservation laws, with application to oil recovery*, Comm. Pure Appl. Math.**40**(1987), no. 2, 141–178. MR**872382**, 10.1002/cpa.3160400202**17.**Zhi Jing Tang and T. C. T. Ting,*Wave curves for the Riemann problem of plane waves in isotropic elastic solids*, Internat. J. Engrg. Sci.**25**(1987), no. 11-12, 1343–1381. MR**921358**, 10.1016/0020-7225(87)90014-0**18.**-,*Solution of a Riemann problem for elasticity*, Journal of Elasticity**26**(1991), 43-63.**19.**B. Wendroff,*The Riemann problem for materials with non-convex equations of state: I, isentropic flow; II, general flow*, J. Math. Anal. Appl.**38**(1972), 454-466; 640-658.

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Additional Information

**Arthur V. Azevedo**

Affiliation:
Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília, DF, Brazil

Email:
arthur@mat.unb.br

**Cesar S. Eschenazi**

Affiliation:
Departamento de Matemática, Universidade Federal de Minas Gerais, Avenida Antônio Carlos 6627, 31270-901 Belo Horizonte, MG, Brazil

Email:
cesar@mat.ufmg.br

**Dan Marchesin**

Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada D. Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil

Email:
marchesi@impa.br

**Carlos F. B. Palmeira**

Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente 225, 22453-900 Rio de Janeiro, RJ, Brazil

Email:
fredpalm@mat.puc-rio.br

DOI:
http://dx.doi.org/10.1090/S0033-569X-10-01154-7

Keywords:
Conservation laws,
Riemann problem,
wave curves,
wave manifold,
intermediate surface

Received by editor(s):
October 23, 2008

Published electronically:
February 19, 2010

Additional Notes:
The first author was supported in part by FEMAT under Grant 04/10, UnB under Grant FUNPE 2005, and CNPq/PADCT under Grant 620029/2004-8

The second author was supported in part by CNPq under Grant 151363/2007-2 and FAPERJ under Grant E-26/150533/2007

The third author was supported in part by CNPq under Grants 472067/2006-0, 304168/2006-8 and 491148/2005-4, and by FAPERJ under Grants E-26/152.525/2006, E-26/110.310/2007.

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Brown University

The copyright for this article reverts to public domain 28 years after publication.