Non-existence and non-uniqueness for multidimensional sticky particle systems

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June 2014, 7(2): 205-218. doi: 10.3934/krm.2014.7.205

Non-existence and non-uniqueness for multidimensional sticky particle systems

Alberto Bressan ^1, and Truyen Nguyen ^2,

1.	Department of Mathematics, Penn State University, University Park, Pa.16802
2.	Department of Mathematics, University of Akron, Akron, OH 44325, United States

Received December 2013 Revised February 2014 Published March 2014

Abstract
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The paper is concerned with sticky weak solutions to the equations of pressureless gases in two or more space dimensions. Various initial data are constructed, showing that the Cauchy problem can have (i) two distinct sticky solutions, or (ii) no sticky solution, not even locally in time. In both cases the initial density is smooth with compact support, while the initial velocity field is continuous.

Keywords: non-uniqueness, Sticky particle systems, non-existence, conservation laws., pressureless Euler system.

Mathematics Subject Classification: Primary: 35L03, 35L65, 35L80; Secondary: 70F1.

Citation: Alberto Bressan, Truyen Nguyen. Non-existence and non-uniqueness for multidimensional sticky particle systems. Kinetic and Related Models, 2014, 7 (2) : 205-218. doi: 10.3934/krm.2014.7.205

References:

[1]	F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Part. Diff. Eq., 24 (1999), 2173-2189. doi: 10.1080/03605309908821498.
[2]	Y. Brenier, W. Gangbo, G. Savaré and M. Westdickenberg, Sticky particle dynamics with interactions, J. Math. Pures Appl., 99 (2013), 577-617. doi: 10.1016/j.matpur.2012.09.013.
[3]	Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353.
[4]	W. E, Yu. Rykov and Y. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), 349-380. doi: 10.1007/BF02101897.
[5]	F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146. doi: 10.1007/s002200100506.
[6]	L. Natile and G. Savaré, A Wasserstein approach to the one-dimensional sticky particle system, SIAM J. Math. Anal., 41 (2009), 1340-1365. doi: 10.1137/090750809.
[7]	T. Nguyen and A. Tudorascu, Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws, SIAM J. Math. Anal., 40 (2008), 754-775. doi: 10.1137/070704459.
[8]	T. Nguyen and A. Tudorascu, One-dimensional pressureless gas systems with/without viscosity, preprint, (2013).
[9]	M. Sever, An existence theorem in the large for zero-pressure gas dynamics, Diff. Integral Equat., 14 (2001), 1077-1092.
[10]	Y. B. Zeldovich, Gravitational instability: An approximate theory for large density perturbations, Astro. & Astrophys., 5 (1970), 84-89.

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References:

[1]	F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Part. Diff. Eq., 24 (1999), 2173-2189. doi: 10.1080/03605309908821498.
[2]	Y. Brenier, W. Gangbo, G. Savaré and M. Westdickenberg, Sticky particle dynamics with interactions, J. Math. Pures Appl., 99 (2013), 577-617. doi: 10.1016/j.matpur.2012.09.013.
[3]	Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353.
[4]	W. E, Yu. Rykov and Y. Sinai, Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., 177 (1996), 349-380. doi: 10.1007/BF02101897.
[5]	F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146. doi: 10.1007/s002200100506.
[6]	L. Natile and G. Savaré, A Wasserstein approach to the one-dimensional sticky particle system, SIAM J. Math. Anal., 41 (2009), 1340-1365. doi: 10.1137/090750809.
[7]	T. Nguyen and A. Tudorascu, Pressureless Euler/Euler-Poisson systems via adhesion dynamics and scalar conservation laws, SIAM J. Math. Anal., 40 (2008), 754-775. doi: 10.1137/070704459.
[8]	T. Nguyen and A. Tudorascu, One-dimensional pressureless gas systems with/without viscosity, preprint, (2013).
[9]	M. Sever, An existence theorem in the large for zero-pressure gas dynamics, Diff. Integral Equat., 14 (2001), 1077-1092.
[10]	Y. B. Zeldovich, Gravitational instability: An approximate theory for large density perturbations, Astro. & Astrophys., 5 (1970), 84-89.

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