On finite-time blowup mechanism of irrotational compressible Euler equations with time-dependent damping. (English) Zbl 1492.35197
Summary: In this paper, sufficient initial conditions for finite-time blowup of smooth solutions of the irrotational compressible Euler equations with time-dependent damping are established. Our blowup conditions reveal that for sufficiently large initial velocity, fixed background density and with no largeness assumption on the initial density, the velocity of the fluid must collapse in finite time on some subset of general Euclidean space with non-zero Lebesgue measure.
MSC:
| 35Q31 | Euler equations |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 35B44 | Blow-up in context of PDEs |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
Keywords:
blowup; time-dependent damping; compressible Euler equations; exponential growth; irrotational fluidsReferences:
| [1] | Lions, PL., Mathematical topics in fluid mechanics, 1 (1996), Oxford: Clarendon Press, Oxford · Zbl 0866.76002 |
| [2] | Lions, PL., Mathematical topics in fluid mechanics, 2 (1998), Oxford: Clarendon Press, Oxford · Zbl 0908.76004 |
| [3] | Sideris, TC., Formation of singularities in three-dimensional compressible fluids, Comm Math Phys, 101, 475-485 (1985) · Zbl 0606.76088 |
| [4] | Sideris, TC., Formation of singularities in solutions to nonlinear hyperbolic equations, Arch Rational Mech Anal, 86, 369-381 (1984) · Zbl 0564.35070 |
| [5] | Cheung, KL., Blowup phenomena for the N-dimensional compressible Euler equations with damping, Z Angew Math Phys, 68, 1-8 (2017) · Zbl 1369.35048 |
| [6] | Cheung, KL., Blowup of solutions for the initial boundary value problem of the 3-dimensional compressible damped Euler equations, Math Method Appl Sci, 41, 4754-4762 (2018) · Zbl 1402.35050 |
| [7] | Hou, F.; Yin, H., On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping, Nonlinearity, 30, 2485-2517 (2017) · Zbl 1373.35229 |
| [8] | Pan, X., Blow up of solutions to 1-Euler equations with time-dependent damping, J Math Anal Appl, 442, 435-445 (2016) · Zbl 1342.35235 |
| [9] | Pan, X., Global existence of solutions to 1-d Euler equations with time-dependent damping, Nonlinear Anal, 132, 327-336 (2016) · Zbl 1328.35155 |
| [10] | Sideris, TC; Thomases, B.; Wang, DH., Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm Partial Differ Equ, 28, 795-816 (2003) · Zbl 1048.35051 |
| [11] | Yuen, MW., Blowup for irrotational \(####\) solutions of the compressible Euler equations in \(####\), Nonlinear Anal, 158, 132-141 (2017) · Zbl 1366.35009 |
| [12] | Beale, JT; Kato, T.; Majda, A., marks on the breakdown of smooth solutions for the 3-D Euler equations, Comm Math Phys, 94, 61-66 (1984) · Zbl 0573.76029 |
| [13] | Elgindi, TM. Finite-time singularity formation for \(####\) solutions to the incompressible Euler equations on \(####\). arXiv:1904.04795v2. · Zbl 1492.35199 |
| [14] | Luk, J.; Speck, J., Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent Math, 214, 1-169 (2018) · Zbl 1409.35142 |
| [15] | Zhu, X., Blowup of the solutions for the IBVP of the isentropic Euler equations with damping, J Math Anal Appl, 432, 715-724 (2015) · Zbl 1325.35154 |
| [16] | Zhu, X.; Tu, A., Blowup of the axis-symmetric solutions for the IBVP of the isentropic Euler equations, Nonlinear Anal, 95, 99-106 (2014) · Zbl 1284.35090 |
| [17] | Wong, S.; Yuen, MW., Blow-up phenomena for compressible Euler equations with non-vacuum initial data, Z Angew Math Phys, 66, 2941-2955 (2015) · Zbl 1327.35308 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
