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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353.  Google Scholar

[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.  Google Scholar

[5]

F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146. doi: 10.1007/s002200100506.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. Nguyen and A. Tudorascu, One-dimensional pressureless gas systems with/without viscosity, preprint, (2013). Google Scholar

[9]

M. Sever, An existence theorem in the large for zero-pressure gas dynamics, Diff. Integral Equat., 14 (2001), 1077-1092.  Google Scholar

[10]

Y. B. Zeldovich, Gravitational instability: An approximate theory for large density perturbations, Astro. & Astrophys., 5 (1970), 84-89. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), 2317-2328. doi: 10.1137/S0036142997317353.  Google Scholar

[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.  Google Scholar

[5]

F. Huang and Z. Wang, Well posedness for pressureless flow, Comm. Math. Phys., 222 (2001), 117-146. doi: 10.1007/s002200100506.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. Nguyen and A. Tudorascu, One-dimensional pressureless gas systems with/without viscosity, preprint, (2013). Google Scholar

[9]

M. Sever, An existence theorem in the large for zero-pressure gas dynamics, Diff. Integral Equat., 14 (2001), 1077-1092.  Google Scholar

[10]

Y. B. Zeldovich, Gravitational instability: An approximate theory for large density perturbations, Astro. & Astrophys., 5 (1970), 84-89. Google Scholar

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