On measure-valued solutions to a two-dimensional gravity-driven avalanche flow model

Mathematical Methods in the Applied Sciences

Volumn 28, Issue 18, 2005, Pages 2201-2223

  Gwiazda, Piotr ^a  

^a UNIVERSITY OF WARSAW   (Poland)

Author keywords

Generalizations of the Young measures; Granular flow; Measure valued solutions; X dependent maximal monotone graph

Indexed keywords

FRICTION; GRANULAR MATERIALS; GRAPH THEORY;

GRANULAR FLOW; MEASURE-VALUE SOLUTIONS; X-DEPENDENT MAXIMAL MONOTONE GRAPH;

FLOW OF FLUIDS;

EID: 29244460994     PISSN: 01704214     EISSN: None     Source Type: Journal    
DOI: 10.1002/mma.660     Document Type: Article

References (39)

1. Savage SB, Hutter K. The motion of a finite mass of granular material down a rough incline. Journal of Fluid Mechanics 1989; 199:177-215.
2. Savage SB, Hutter K. The dynamics of avalanches of granular materials from initiation to runout. I. Analysis. Acta Mechanica 1991; 86(1-4):201-223.
3. Gwiazda P. An existence result for a model of granular material with non-constant density. Asymptotic Analysis 2002; 30(1):43-60.
4. Bouchut F, Westdickenberg M. Gravity driven shallow water models for arbitrary topography. Communications in Mathematical Sciences 2004; 2(3):359-389.
5. Bouchut F, Mangeney-Castelnau A, Perthame B, Vilotte J-P. A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows. Comptes Rendus de l Mathematique Academie des Sciences, Paris 2003; 336(6):531-536.
6. Gray JMNT, Wieland M, Hutter K. Gravity-driven free surface flow of granular avalanches over complex basal topography. Proceedings of the Royal Society of London, Series A-Mathematical, Physical and Engineering Sciences 1999; 455(1985):1841-1874.
7. Wieland M, Gray JMNT, Hutter K. Channelized free-surface flown of cohesionless granular avalanches in a chute with shallow lateral curvature. Journal of Fluid Mechanics 1999; 392:73-100.
8. Gray JMNT, Tai Y-C, Noelle S. Shock waves, dead zones and particle-free regions in rapid granular free-surface flows. Journal of Fluid Mechanics 2003; 491:161-181.
9. Farwig R. Existence of avalanching flows. In Dynamic Response of Granular and Porous Materials under Larae and Catastrophic Deformations, Kirchner N, Hutter K (eds). Lecture Notes in Applied and Computational Mechanics, vol. 11. Springer: Berlin, Heidelberg, 2003.
10. Pouliquen O, Forterre Y. Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane. Journal of Fluid Mechanics 2002; 453:133-151.
11. 1-stability and uniqueness result for balance laws with multifunctions: a model form the theory of granular media. Colloquium Mathematicum 2004; 100(2): 149-162.
12. Lions P-L, Perthame B, Souganidis PE. Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Communications on Pure and Applied Mathematics 1996; 49(6):599-638.
13. Lions P-L. Mathematical Topics in Fluid Mechanics. Oxford Lecture Series in Mathematics and its Applications, vols. 2,10. The Clarendon Press, Oxford University Press: New York, 1998. Compressible Models. Oxford Science Publications: Oxford.
14. Perthame B. Kinetic Formulation of Conservation Laws. Oxford Lecture Series in Mathematics and its Applications, vol. 21. Oxford University Press: Oxford, 2002.
15. DiPerna RJ. Measure-valued solutions to conservation laws. Archive for Rational Mechanics and Analysis 1985; 88(3):223-270.
16. DiPema RJ, Majda AJ. Oscillations and concentrations in weak solutions of the incompressible fluid equations. Communications in Mathematical Physics 1987; 108(4):667-689.
17. DiPerna RJ, Majda AJ. Concentrations in regularizations for 2-D incompressible flow. Communications on Pure and Applied Mathematics 1987; 40(3):301-345.
18. DiPema RJ, Majda A. Reduced Hausdorff dimension and concentration-cancellation for two-dimensional incompressible flow. Journal of the American Mathematical Society 1988; 1(1):59-95.
19. Málek J, Nečas J, Rokyta M, Růžička M. Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation, vol. 13. Chapman & Hall: London, Weinheim, 1996.
20. Kröener D, Zajaczkowski WM. Measure-valued solutions of the Euler equations for ideal compressible polytropic fluids. Mathematical Methods in the Applied Sciences 1996; 19(3):235-252.
21. Ball JM. A version of the fundamental theorem for Young measures. PDEs and Continuum Models of Phase Transitions (Nice, 1988). Lecture Notes in Physics, vol. 344. Springer: Berlin, 1989; 207-215.
22. Ball JM, Murat F. Remarks on Chacon's biting lemma. Proceedings of the American Mathematical Society 1989; 107(3):655-663.
23. Alibert JJ, Bouchitté G. Non-uniform integrability and generalized Young measures. Journal of Convex Analysis 1997; 4(1):129-147.
24. Francfort G, Murat F, Tatar L. Monotone operators in divergence form with x-dependent multivalued graphs. Bollettino delta Unione Matematica Italiana Sez. B Artic. Ric. Mat. (8) 2004; 7(1):23-59.
25. n. Mathematische Zeitschrift 1999; 230(2):259-316.
26. Kałemajska A. Between the classical theorem of Young and convergence theorem in set-valued analysis. Instytut Mat. PAN, 635, 2003, preprint.
27. Gwiazda P, Zatorska-Goldstein A. Existence via compactness for maximal monotone elliptic operators. Comptes Rendus de l Mathématique Academie des Sciences, Paris 2005; 340(7):489-492.
28. Gwiazda P, Málek J, Świerczewska A. On flows of an incompressible fluid with discontinuous power-law-like rheology. École Normale Supérieure, 05-04, 2005, preprint.
29. Gwiazda P. An existence result for balance laws with multifunctions: a model from the theory of granular media. Colloquium Mathematicum 2003; 97(1):67-79.
30. Balder EJ. Lectures on Young measure theory and its applications to economics. Rendiconti dell' Istituto di Matematica dell' Università di Trieste 2000; 31(Suppl. 1):1-69. Workshop on Measure Theory and Real Analysis (Italian), Grado, 1997.
31. Hungerbühler N. A refinement of Ball's theorem on Young measures. New York Journal of Mathematics 1997; 3:48-53 (electronic).
32. Müller S. Variational models for microstructure and phase transitions. In Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), Hildebrandt S, Struwe M (eds), Lectures Notes in Mathematics, vol. 1713. Springer: Berlin, Heidelberg, 1999; 85-210.
33. Roubíček T. Relaxation in Optimization Theory and Variational Calculus, de Gruyter Series in Nonlinear Analysis and Applications, vol. 4. Walter de Gruyter & Co.: Berlin, 1997.
34. DiPerna RJ, Lions P-L. Ordinary differential equations, transport theory and Sobolev spaces. Inventiones Mathematicae 1989; 98(3):511-547.
35. Ambrosio L. Transport equation and Cauchy problem for BV vector fields. Inventiones Mathematicae 2004; 158(2):227-260.
36. Federer H. Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer: New York, 1969.
37. Minty GJ. Monotone (nonlinear) operators in Hubert space. Duke Mathematical Journal 1962; 29:341-346.
38. Aubin J-P, Frankowska H. Set-Valued Analysis. Systems & Control: Foundations & Applications, vol. 2. Birkhäuser: Boston, MA, 1990.
39. Piat VC, Maso GD, Defrancheschi A. G-convergence of monotone operators. Annales de I Institut Henri Poincaré - Analyse Non Linéaire 1990; 7(3):123-160.