Euler equation existence, non-uniqueness and mesh converged statistics. (English) Zbl 1353.35227
Summary: We review existence and non-uniqueness results for the Euler equation of fluid flow. These results are placed in the context of physical models and their solutions. Non-uniqueness is in direct conflict with the purpose of practical simulations, so that a mitigating strategy, outlined here, is important. We illustrate these issues in an examination of mesh converged turbulent statistics, with comparison to laboratory experiments.
MSC:
| 35Q31 | Euler equations |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
References:
| [1] | Ball JM . 1989 A version of the fundamental theorem of Young measures. In PDEs and continuum models of phase transitions. Lecture Notes in Physics, vol. 344, pp. 207–215. New York, NY: Springer. · doi:10.1007/BFb0024945 |
| [2] | Neuman SP . 1993 Eulerian–Langrangian theory of transport in space–time nonstationary velocity fields: exact nonlocal formalism by conditional moments and weak approximation. Water Resour. Res. 29, 633–645. (doi:10.1029/92WR02306) · doi:10.1029/92WR02306 |
| [3] | Gangbo W Westerdickenberg M . 2009 Optimal transport for the system of isentropic Euler equations. Commun. Partial Differ. Equ. 34, 1041–1073. (doi:10.1080/03605300902892345) · Zbl 1182.35161 · doi:10.1080/03605300902892345 |
| [4] | Chen GQ Glimm J . 2012 Kolmogorov’s theory of turbulence and inviscid limit of the Navier–Stokes equations in R 3 . Commun. Math. Phys. 310, 267–283. (doi:10.1007/s00220-011-1404-9) · Zbl 1232.35114 · doi:10.1007/s00220-011-1404-9 |
| [5] | Scheffer V . 1993 An inviscid flow with compact support in space–time. J. Geom. Anal. 3, 343–401. (doi:10.1007/BF02921318) · Zbl 0836.76017 · doi:10.1007/BF02921318 |
| [6] | Shnirelman A . 1997 On the nonuniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50, 1261–1286. (doi:10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6) · Zbl 0909.35109 · doi:10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6 |
| [7] | Leliss CD Szekelyhidi L . 2012 The h-principle and the equations of fluid dynamics. Bull. Amer. Math. Soc. 49, 347–375. (doi:10.1090/S0273-0979-2012-01376-9) · Zbl 1254.35180 · doi:10.1090/S0273-0979-2012-01376-9 |
| [8] | Leonard A . 1975 Energy cascade in large-eddy simulations of turbulent fluid flows. Adv. Geophys. 18, Part A, 237–248. (doi:10.1016/S0065-2687(08)60464-1) |
| [9] | DOI: 10.1063/1.858164 · Zbl 0753.76074 · doi:10.1063/1.858164 |
| [10] | Ma T . 2006 Large eddy simulation of variable density flows. PhD thesis, University of Maryland, College Park, MD, USA. |
| [11] | DOI: 10.1073/pnas.1002410107 · Zbl 1205.76117 · doi:10.1073/pnas.1002410107 |
| [12] | DOI: 10.1098/rsta.2012.0183 · Zbl 1353.76027 · doi:10.1098/rsta.2012.0183 |
| [13] | DOI: 10.1063/1.1688328 · Zbl 1186.76143 · doi:10.1063/1.1688328 |
| [14] | DOI: 10.1063/1.3064120 · Zbl 1183.76366 · doi:10.1063/1.3064120 |
| [15] | Lim H Yu Y Glimm J Sharp DH . 2010 Nearly discontinuous chaotic mixing. High Energy Density Phys. 6, 223–226. (doi:10.1016/j.hedp.2010.01.005) · doi:10.1016/j.hedp.2010.01.005 |
| [16] | DOI: 10.5488/CMP.13.43401 · doi:10.5488/CMP.13.43401 |
| [17] | Kaman T Kaufman R Glimm J Sharp DH . 2012 Uncertainty quantification for turbulent mixing flows: Rayleigh–Taylor instability. In Uncertainty quantification in scientific computing: IFIP advances in information and communication technology, vol. 377, pp. 212–225 (Stony Brook University Preprint number SUNYSB-AMS-11-08). · Zbl 1426.76180 |
| [18] | Kaman T . 2011 A numerical method for the simulation of turbulent mixing and its basis in mathematical theory. In Theory and applications: short course book, pp. 105–129. Lecture Notes on Numerical Methods for Hyperbolic Equations. London, UK: CRC Press. · Zbl 1312.76018 |
| [19] | Melvin J Rao P Kaufman R Lim H Yu Y Glimm J Sharp DH . 2013 Atomic scale mixing for inertial confinement fusion associated hydro instabilities. High Energy Density Phys. 9, 288–296. (doi:10.1016/j.hedp.2013.01.007) · doi:10.1016/j.hedp.2013.01.007 |
| [20] | Melvin J Rao P Kaufman R Lim H Yu Y Glimm J Sharp DH . 2014 Turbulent transport at high Reynolds numbers in an inertial confinement fusion context. J. Fluids Eng. 136, 091206. (doi:10.1115/1.4027382) · doi:10.1115/1.4027382 |
| [21] | Jiao X Zha H . 2008 Consistent computation of first- and second-order differential quantities for surface meshes. In Proc. ACM Symp. on Solid and Physical Modeling, Stony Brook, NY, USA, 2–4 June 2008, pp. 159–170. New York, NY: ACM. () · doi:10.1145/1364901.1364924 |
| [22] | Ray N Wang D Jiao X Glimm J . 2012 High-order numerical integration over discrete surfaces. SIAM J. Numer. Anal. 50, 3061–3083. (doi:10.1137/110857404) · Zbl 1261.65028 · doi:10.1137/110857404 |
| [23] | Clark B Ray N Jiao X . 2013 Surface mesh optimization, adaption and untangling with high-order accuracy. In Proc. 21st International Meshing Roundtable (eds X Jiao, J-C Weill), pp. 385–402. Berlin, Germany: Springer. () · doi:10.1007/978-3-642-33573-0_23 |
| [24] | Kaufman R . 2014 Software tools for stochastic simulations of turbulence. PhD thesis, Stony Brook University, Stony Brook, NY, USA. |
| [25] | DOI: 10.1016/0021-9991(81)90144-3 · Zbl 0469.76079 · doi:10.1016/0021-9991(81)90144-3 |
| [26] | Glimm J Li X Liu Y Xu Z Zhao N . 2003 Conservative front tracking with improved accuracy. SIAM J. Numer. Anal. 41, 1926–1947. (doi:10.1137/S0036142901388627) · Zbl 1053.35093 · doi:10.1137/S0036142901388627 |
| [27] | Liu J Lim H Glimm J Li X . 2007 A conservative front tracking method in N-dimensions. J. Sci. Comput. 31, 213–236. (doi:10.1007/s10915-006-9117-5) · Zbl 1115.76357 · doi:10.1007/s10915-006-9117-5 |
| [28] | Glimm J Plohr B Sharp DH . 2013 Large eddy simulation, turbulent transport and the renormalization group. (http://arxiv.org/abs/1308.3221). · Zbl 1395.76030 |
| [29] | Bird R Stewart W Lightfoot E . 2001 Transport phenomena, 2nd edn. New York, NY: John Wiley & Sons. |
| [30] | Fedkiw RP . 1997 A survey of chemically reacting compressible flows. PhD thesis, University of California Los Angeles, Los Angeles, CA, USA. |
| [31] | Agafontsev DS Kuznetsov EA Mailybaev AA . 2015 Development of high vorticity structures in incompressible 3D Euler equations. (http://arxiv.org/abs/1502.01562) · Zbl 1326.65136 |
| [32] | Mueschke N . 2008 Experimental and numerical study of molecular mixing dynamics in Rayleigh–Taylor unstable flows. PhD thesis, Texas A&M University, College Station, TX, USA. |
| [33] | Melvin J Kaufman R Lim H Kaman T Rao P Glimm J . 2013 Macro and micro issues in turbulent mixing. Sci. China Technol. Sci. 56, 2355–2360. (doi:10.1007/s11431-013-5340-0) · doi:10.1007/s11431-013-5340-0 |
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.
