Classical solutions for the Euler equations of compressible fluid dynamics: A new topological approach

• Boureni, Dalila ^[2] ; Georgiev, Svetlin ^[1] ; Kheloufi, Arezki ^[2] ; Mebarki, Karima ^[2]
  1. [1] Sofia University

     Sofia University

     Bulgaria

     
  2. [2] Bejaia University
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 23, Nº. 2, 2022, págs. 463-480
• Idioma: inglés
• DOI: 10.4995/agt.2022.15963
• Enlaces
  • Texto completo
• Resumen

  • In this article we study a class of Euler equations of compressible fluid dynamics. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results we propose a new approach  based upon  recent  theoretical results.

• Referencias bibliográficas
  • S. Benslimane, S. Georgiev and K. Mebarki, Multiple nonnegative solutions for a class of fourth-order BVPs via a new topological approach,...
  • D. Chae, On the well-posedness of the Euler equations in the Triebel-Lizorkin spaces, Commun. Pure Appl. Math. 55 (2002), 654-78. https://doi.org/10.1002/cpa.10029
  • D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal. 38 (2004), 339-358.
  • G. Q. Chen and J. G. Liu, Convergence of second-order schemes for isentropic gas dynamics, Mathematics of computation 61, no. 204 (1993),...
  • K. L. Cheung and S. Wong, The lifespan of classical solutions to the (damped) compressible Euler equations, Bull. Malays. Math. Sci. Soc....
  • D. Christodoulou, The Euler equations of compressible fluid flow, Bulletin of the American Mathematical Society 44, no. 4 (2007), 581-602....
  • R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Commun. Math. Phys. 91 (1983), 1-30. https://doi.org/10.1007/BF01206047
  • S. Djebali and K. Mebarki, Fixed point index theory for perturbation of expansive mappings by k-set contractions, Top. Meth. Nonli. Anal....
  • P. Drabek and J. Milota, Methods in Nonlinear Analysis, Applications to Differential Equations, Birkhäuser, 2007.
  • D. G. Ebin, Motion of a slightly compressible fluid, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 539-542. https://doi.org/10.1073/pnas.72.2.539
  • S. Georgiev and P. LeFloch, Generalized time-periodic solutions to the Euler equations of compressible fluids, Differ. Equ. Appl. 1, no. 3...
  • J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. https://doi.org/10.1002/cpa.3160180408
  • Y. Goncharov, On existence and uniqueness of classical solutions to Euler equations in a rotating cylinder, Eur. J. Mech. B Fluids 25 (2006),...
  • D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, Mass, USA, vol. 5, (1988).
  • H. Jia and R. Wan, Long time existence of classical solutions for the rotating Euler equations and related models in the optimal Sobolev space,...
  • E. I. Kapstov and S. V. Meleshko, Conservation laws of the one-dimensional isentropic gas dynamics equations in Lagrangian coordinates, AIP...
  • T. Kato, Nonstationary flows of viscous and ideal fluids in $R^{3},$ J. Functional Analysis 9 (1972), 296-305. https://doi.org/10.1016/0022-1236(72)90003-1
  • T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891-907. https://doi.org/10.1002/cpa.3160410704
  • P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, Commun. Math. Phys. 163 (1994),...
  • T. P. Liu and J. A. Smoller, On the vacuum state for isentropic gas dynamics equations, Advances in Applied Mathematics 1 (1980), 345-359....
  • A. Majda, Compressible Fluid Flow and Conservation Laws in Several Space Variables, Springer: Berlin, New York, 1984. https://doi.org/10.1007/978-1-4612-1116-7
  • T. Makino, S. Ukai and S. Kawashima, Sur la solution à support compact de l'équation d'Euler compressible, Japan J. Appl. Math. 3...
  • H. C. Pak and Y. J. Park, Existence of solution for the Euler equations in a critical Besov space $B^{1}_{infty;1}(R^{n}),$ Comm. Partial...
  • X. Pan, Global existence and asymptotic behavior of solutions to the Euler equations with time-dependent damping, Appl. Anal. 100 (2021),...
  • A. Polyanin and A. Manzhirov, Handbook of Integral Equations, CRC Press, 1998. https://doi.org/10.1201/9781420050066
  • D. Serre, Solutions classiques globales des équations d'Euler pour un fluide parfait compressible, Ann. Inst. Fourier (Grenoble) 47, no....
  • T. C. Sideris, B. Thomases and D. Dehua Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial...
  • R. Takada, Long time existence of classical solutions for the 3D incompressible rotating Euler equations, J. Math. Soc. Japan 68, no. 2 (2016),...
  • T Xiang and R. Yuan, A class of expansive-type Krasnosel'skii fixed point theorems, Nonlinear Anal. 71, no. 7-8 (2009), 3229-3239. https://doi.org/10.1016/j.na.2009.01.197
  • J. Xu and S. Kawashima, Diffusive relaxation limit of classical solutions to the damped compressible Euler equations, J. Differential Equations...
  • N. Zabusky, Fermi-Pasta-Ulam, solitons and the fabric of nonlinear and computational science: history, synergetics, and visiometrics, Chaos...