The inviscid, compressible and rotational, 2D isotropic Burgers and pressureless Euler-Coriolis fluids: solvable models with illustrations.

*(English)*Zbl 1349.76041Summary: The coupling between dilatation and vorticity, two coexisting and fundamental processes in fluid dynamics [J.-Z. Wu, H.-X. Ma and M.-D. Zhou, Vorticity and vortex dynamics. Berlin etc.: Springer-Verlag (2006; doi:10.1007/978-3-540-29028-5)] is investigated here, in the simplest cases of inviscid 2D isotropic Burgers and pressureless Euler-Coriolis fluids respectively modeled by single vortices confined in compressible, local, inertial and global, rotating, environments. The field equations are established, inductively, starting from the equations of the characteristics solved with an initial Helmholtz decomposition of the velocity fields namely a vorticity free and a divergence free part [loc. cit., Sections 2.3.2, 2.3.3) and, deductively, by means of a canonical Hamiltonian A. Clebsch like formalism [J. Reine Angew. Math. 54, 293–312 (1857; ERAM 054.1440cj); ibid. 56, 1–10 (1859; ERAM 056.1468cj)], implying two pairs of conjugate variables. Two vector valued fields are constants of the motion: the velocity field in the Burgers case and the momentum field per unit mass in the Euler-Coriolis one. Taking advantage of this property, a class of solutions for the mass densities of the fluids is given by the Jacobian of their sum with respect to the actual coordinates. Implementation of the isotropy hypothesis entails a radial dependence of the velocity potentials and of the stream functions associated to the compressible and to the rotational part of the fluids and results in the cancellation of the dilatation-rotational cross terms in the Jacobian. A simple expression is obtained for all the radially symmetric Jacobians occurring in the theory. Representative examples of regular and singular solutions are shown and the competition between dilatation and vorticity is illustrated. Inspired by thermodynamical, mean field theoretical analogies, a genuine variational formula is proposed which yields unique measure solutions for the radially symmetric fluid densities investigated. We stress that this variational formula, unlike the Hopf-Lax formula, enables us to treat systems which are both compressible and rotational. Moreover in the one-dimensional case, we show for an interesting application that both variational formulas are equivalent.

##### MSC:

76B47 | Vortex flows for incompressible inviscid fluids |

76M30 | Variational methods applied to problems in fluid mechanics |

##### Keywords:

cylindrical vortices; compressible and rotational; Maxwell construction; weak solutions; variational formula
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\textit{Ph. Choquard} and \textit{M. Vuffray}, Physica D 285, 18--27 (2014; Zbl 1349.76041)

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##### References:

[1] | Wu, J.-Z.; Ma, H.-Y.; Zhou, J., Vorticity and vortex dynamics, (2006), Springer |

[2] | Lions, P.-L., Mathematical topics in fluid mechanics: volume 1: incompressible models, (1996), Oxford University Press |

[3] | Majda, A.; Bertozzi, A.; Ogawa, A., Vorticity and incompressible flow. Cambridge texts in applied mathematics, Appl. Mech. Rev., 55, 77, (2002) |

[4] | Marchioro, C.; Pulvirenti, M., Mathematical theory of incompressible nonviscous fluids, vol. 96, (1994), Springer · Zbl 0789.76002 |

[5] | Shnirelman, A., Weak solutions of incompressible Euler equations, (Handbook of Mathematical Fluid Dynamics, vol. 2, (2003), North-Holland), 87-116, (Chapter 3) · Zbl 1143.76394 |

[6] | Lions, P.-L., Mathematical topics in fluid mechanics: volume 2: compressible models, (1998), Oxford University Press |

[7] | Chen, G.-Q.; Wang, D., The Cauchy problem for the Euler equations for compressible fluids, (Handbook of Mathematical Fluid Dynamics, vol. 1, (2002), North-Holland), 421-543, (Chapter 5) · Zbl 1230.35096 |

[8] | L. Yanguang Charles, Integrable structures for 2D Euler equations of incompressible inviscid fluids, in: Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 43, 2002, pp. 332-338. · Zbl 1092.37056 |

[9] | Yuen, M., Some exact blowup solutions to the pressureless Euler equations in RN, Commun. Nonlinear Sci. Numer. Simul., 16, 8, 2993-2998, (2011) · Zbl 1419.76542 |

[10] | Evans, L. C., Partial differential equations, (1998), American Mathematical Society Providence, Rhode Land |

[11] | Clebsch, A., Über eine allgemeine transformation der hydrodynamischen gleichungen, J. Reine Angew. Math., 54, 293-312, (1857) |

[12] | Clebsch, A., Ueber die integration der hydrodynamischen gleichungen, J. Reine Angew. Math., 56, 1-10, (1859) |

[13] | Choquard, P., Single speed solutions of the Vlasov-Poisson equations for coulombian and Newtonian systems in nd, Commun. Nonlinear Sci. Numer. Simul., 13, 1, 40-45, (2008), Vlasovia 2006: The Second International Workshop on the Theory and Applications of the Vlasov Equation · Zbl 1130.35365 |

[14] | Choquard, P., On a class of mean field solutions of the Monge problem for perfect and self-interacting systems, Transport Theory Statist. Phys., 39, 5-7, 313-359, (2010) · Zbl 1222.82038 |

[15] | Morrison, P., Hamiltonian fluid dynamics, (Françoise, J.-P.; Naber, G. L.; Tsun, T. S., Encyclopedia of Mathematical Physics, (2006), Academic Press Oxford), 593-600 |

[16] | Gel’fand, I. M., Some problems in the theory of quasi-linear equations, Uspekhi Mat. Nauk, 14, 2, 87-158, (1959) · Zbl 0096.06602 |

[17] | Choquard, P.; Wagner, J., On the “mean field” interpretation of burgers’ equation, J. Stat. Phys., 116, 1-4, 843-853, (2004) · Zbl 1142.82380 |

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