We provide the breakdown mechanism of pressureless gases when the initial vorticity is zero. In other words, the maximum norm of the divergence and $\Vert u\Vert_{H^{\frac{1}{2}}}$ control the breakdown of the solution. Then we show that the solution must blow up for certain initial data in both non-relativistic and relativistic settings.
Wei HUO
. ON SINGULARITY FORMATION FOR 3-$D$ ZERO PRESSURE FLOW[J]. Acta mathematica scientia, Series B, 2025
, 45(3)
: 937
-950
.
DOI: 10.1007/s10473-025-0311-7
[1] Bahouri H, Chemin J Y, Danchin R.Fourier Analysis and Nonlinear Partial Differential Equations. Heidelberg: Springer, 2011
[2] Beale J T, Kato T, Majda A. Remarks on the breakdown of smooth solutions for the 3-$D$ Euler equations. Comm Math Phys, 1984, 94(1): 61-66
[3] Boudin L. A solution with bounded expansion rate to the model of viscous pressureless gases. SIAM J Math Anal, 2000, 32(1): 172-193
[4] Brenier Y, Grenier E. Sticky particles and scalar conservation laws. SIAM J Num Anal, 1998, 35(6): 2317-2328
[5] Carnevale G F, Pomeau Y, Young W R. Statistics of ballistic agglomeration. Phys Rev Lett, 1990, 64(24): 2913-2916
[6] Carrillo J A, Wróblewska-Kamińska A, Zatorska E. Pressureless Euler with nonlocal interactions as a singular limit of degenerate Navier-Stokes system. J Math Anal Appl, 2020, 492(1): 124400
[7] Cavalletti F, Sedjro M, Westdickenberg M. A variational time discretization for compressible Euler equations. Trans Amer Math Soc, 2019, 371(7): 5083-5155
[8] Christodoulou D. Self-gravitating relativistic fluids: A two-phase model. Arch Rational Mech Anal, 1995, 130(4): 343-400
[9] Danchin R. Global solutions for two-dimensional viscous pressureless flows with large variations of density. Probab Math Phys, 2024, 5(1): 55-88
[10] Danchin R, Mucha P B, Tolksdorf P. Lorentz spaces in action on pressureless systems arising from models of collective behavior. J Evol Equ, 2021, 21(3): 3103-3127
[11] Dermoune A. {$d$}-dimensional pressureless gas equations. Theory Probab Appl, 2005, 49(3): 540-545
[12] E W, Rykov Y G, Sinai Y G. 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, 1996, 177(2): 349-380
[13] Huang F. Weak solution to pressureless type system. Comm Partial Differential Equations, 2005, 30: 283-304
[14] Huang F, Wang Z. Well posedness for pressureless flow. Comm Math Phys, 2001, 222: 117-146
[15] Landau L D, Lifshitz E M. Fluid Mechanics.New York: Pergamon Press, 1987
[16] Lax P D.Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. Philadelphia: SIAM, 1973
[17] Majda A.Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. New York: Springer-Verlag, 1984
[18] Mucha P B, Ożański W S. Global-in-time stability of ground states of a pressureless hydrodynamic model of collective behaviour. Commun Math Sci, 2023, 21(7): 1937-1959
[19] Shandarin S F, Zeldovich Y B. The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium. Rev Mod Phys, 1989, 61(2): 185-220
[20] Shen C, Sun M. Exact Riemann solutions for the drift-flux equations of two-phase flow under gravity. J Differ Equ, 2022, 314: 1-55
[21] Taylor M E.Partial Differential Equations III: Nonlinear Equations. Corrected reprint of the 1996 original. New York: Springer-Verlag, 1997
[22] Yang H, Zhang Y. Flux approximation to the isentropic relativistic Euler equations. Nonlinear Anal, 2016, 133: 200-227
[23] Yang H, Liu J. Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation. Sci China Math, 2015, 58(11), 2329-2346
[24] Zeldovich Y B. Gravitational instability: An approximate theory for large density perturbations. Astron Astrophys, 1970, 5: 84-89
