{
  "id": "1402.0487",
  "version": "v4",
  "published": "2014-02-03T19:57:55.000Z",
  "updated": "2015-04-15T16:47:07.000Z",
  "title": "Large order Reynolds expansions for the Navier-Stokes equations",
  "authors": [
    "Carlo Morosi",
    "Mario Pernici",
    "Livio Pizzocchero"
  ],
  "comment": "Some overlaps with our works arXiv:1405.3421, arXiv:1310.5642, arXiv:1304.2972, arXiv:1203.6865, arXiv:1104.3832, arXiv:1009.2051, arXiv:1007.4412, arXiv:0909.3707, arXiv:0709.1670. These overlaps aim to make the paper self-cointained and do not involve the main results",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Cauchy problem for the incompressible homogeneous Navier-Stokes (NS) equations on a d-dimensional torus, in the C^infinity formulation described, e.g., in [25]. In [22][25] it was shown how to obtain quantitative estimates on the exact solution of the NS Cauchy problem via the \"a posteriori\" analysis of an approximate solution; such estimates concern the interval of existence of the exact solution and its distance from the approximate solution. In the present paper we consider an approximate solutions of the NS Cauchy problem having the form u^N(t) = sum_{j=0}^N R^j u_j(t), where R is the \"mathematical\" Reynolds number (the reciprocal of the kinematic viscosity) and the coefficients u_j(t) are determined stipulating that the NS equations be satisfied up to an error O(R^{N+1}). This subject was already treated in [24], where, as an application, the Reynolds expansion of order N=5 in dimension d=3 was considered for the initial datum of Behr-Necas-Wu (BNW). In the present paper, these results are enriched regarding both the theoretical analysis and the applications. Concerning the theoretical aspect, we refine the approach of [24] following [25] and use the symmetries of the initial datum in building up the expansion. Concerning the applicative aspect we consider two more (d=3) initial data, namely, the vortices of Taylor-Green (TG) and Kida-Murakami (KM); the Reynolds expansions for the BNW, TG and KM data are performed via a Python program, attaining orders between N=12 and N=20. Our a posteriori analysis proves, amongst else, that the solution of the NS equations with anyone of the above three data is global if R is below an explicitly computed critical value. Our critical Reynolds numbers are below the ones characterizing the turbulent regime; however these bounds have a sound theoretical support, are fully quantitative and improve previous results of global existence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-03T19:57:55.000Z",
      "abstract": "We consider the incompressible homogeneous Navier-Stokes (NS) equations on T^d = (R/2 pi Z)^d, in the setting of the Sobolev spaces H^n(T^d) of divergence free, zero mean vector fields (n > d/2+1). Morosi and Pizzocchero [16] treated the approximate solutions of the NS Cauchy problem having the form u^N(t) = sum_{j=0}^N R^j u_j(t), where R is the \"mathematical\" Reynolds number (the reciprocal of the kinematic viscosity) and the coefficients u_j(t) are determined stipulating that the NS equations be satisfied up to an error O(R^{N+1}). In the cited work it was shown how to obtain quantitative estimates on the exact solution u of the NS Cauchy problem via a posteriori analysis of the Reynolds expansion u^N; such estimates concern the interval of existence of u and the Sobolev distance between u(t) and u^N(t). This framework was exemplified in dimension d=3 with the initial datum of Behr, Necas and Wu, computing via Mathematica the Reynolds expansion up to the order N=5. The present work contains results on this subject obtained by a more efficient Python program; this has allowed us to push the Reynolds expansion up to the order N=20 for the Behr-Necas-Wu (BNW) datum. We have considered two more (d=3) initial data, namely, the vortices of Taylor-Green (TG) and Kida-Murakami (KM); the expansions for these data have been built via Python up to the orders N=20 and N=12, respectively. Our analysis grants, amongst else, that the solution of the NS equations with the three data mentioned above is global if R is below a critical value. This result can be reformulated in terms of the \"physical\" Reynolds number Re := V_{*} L_{*} R, where V_{*} is the initial mean quadratic velocity and L_{*} is the mean quadratic wavelength of the initial datum. The conclusion is that we have global existence for the BNW, TG and KM initial data if Re <= 7.84, Re <= 5.07 and Re <= 1.00, respectively.",
      "comment": "Some overlaps with our previous works arXiv:1310.5642, arXiv:1304.2972, arXiv:1203.6865, arXiv:1104.3832, arXiv:1009.2051, arXiv:1007.4412, arXiv:0909.3707, arXiv:0709.1670. These overlaps aim to make the paper self-cointained and do not involve the main results which are: i) an improved Reynolds expansion for the BNW datum; ii) new Reynolds expansions for the TG and KM data",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-20T17:23:17.000Z",
      "abstract": "We consider the Cauchy problem for the incompressible homogeneous Navier-Stokes (NS) equations on a d-dimensional torus (typically, with d=3), in the C^infinity formulation described, e.g., in [Morosi and Pizzocchero, 2015]. In the cited work and in [Morosi and Pizzocchero, 2012] it was shown how to obtain quantitative estimates on the exact solution of the NS Cauchy problem via the a posteriori analysis of an approximate solution; such estimates concern the interval of existence of the exact solution and its distance from the approximate solution, evaluated in terms of Sobolev norms. In the present paper we consider an approximate solutions of the NS Cauchy problem having the form u^N(t) = sum_{j=0}^N R^j u_j(t), where R is the \"mathematical\" Reynolds number (the reciprocal of the kinematic viscosity) and the coefficients u_j(t) are determined stipulating that the NS equations be satisfied up to an error O(R^{N+1}). This subject was already treated in [Morosi and Pizzocchero, 2014], where, as an application, we considered the Reynolds expansion of order N=5 in dimension d=3, choosing for the initial datum the Behr-Necas-Wu vortex. In the present paper, these results are enriched regarding both the theoretical analysis and the applications. Concerning the second aspect we consider two more (d=3) initial data, namely, the vortices of Taylor-Green and Kida-Murakami; the Reynolds expansions for the above three vortices are performed symbolically via a Python program, attaining orders between N=12 and N=20. Our a posteriori analysis proves, amongst else, that the solution of the NS equations with anyone of the above three data is global if R is below an explicitly computed critical value. Admittedly, our critical Reynolds numbers are below the ones characterizing the turbulent regime; however these bounds are rigorous, fully quantitative and improve previous results of global existence.",
      "comment": "Some overlaps with our works arXiv:1405.3421, arXiv:1310.5642, arXiv:1304.2972, arXiv:1203.6865, arXiv:1104.3832, arXiv:1009.2051, arXiv:1007.4412, arXiv:0909.3707, arXiv:0709.1670. These overlaps aim to make the paper self-cointained and do not involve the main results which are: i) an improved Reynolds expansion for the BNW datum; ii) new Reynolds expansions for the TG and KM data",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-04-15T16:47:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "76D03",
      "76D05"
    ],
    "keywords": [
      "large order reynolds expansions",
      "navier-stokes equations",
      "initial datum",
      "ns cauchy problem",
      "initial mean quadratic velocity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.0487M"
    }
  }
}