Well-posedness for the diffusive 3D Burgers equations with initial data in H½

Abstract

This material has been published in Recent Progress in the Theory of the Euler and Navier-Stokes Equations edited by Robinson, James C. and Rodrigo, José L. and Sadowski, Witold, et al. This version is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription.

The Navier–Stokes regularity problem

“…T 0 ( Ω |u(x, t)| s dx) r/s dt < ∞, with 2/r + 3/s = 1 then u is smooth. 9 The proof of this result follows similar lines to that above, only using some more refined inequalities along the way. The proof of the other endpoint case (r = ∞, s = 3), when |u| 3 is bounded in time, is very involved and was only given in 2003 by Escauriaza et al [31].…”

“…There are artificial models that share many of the features of the Navier-Stokes equations that have regular solutions (e.g. the dissipative Burgers equation [9]) and other such models that have solutions that blow up in finite time [10,11]. It may be that an eventual proof/disproof of the regularity of solutions of the Navier-Stokes equations will involve such particular features of the equations that it says little about other models.…”

“…These results are valid for a weak solutions that also satisfy a localized version of the energy inequality, termed 'suitable weak solutions' in [17]. The 9 For a solution u with T 0 Ω |u(x, t)| 4 < ∞, Prodi [27] proved uniqueness under this condition; in [22] Serrin showed that under this condition alone a solution u is unique in the class of all weak solutions that satisfy the energy inequality. In [28], Serrin proved a local conditional regularity result: if u ∈ L r (t 1 , t 2 ; L s (U)) for some 2/r + 3/s < 1 then u is smooth in space in the space-time domain (t 1 , t 2 ) × U; the proof of this result is much harder (and that in the case 2/r + 3/s = 1, due to Fabes et al [29] and Struwe [30], harder still).…”

On a model for the Navier–Stokes equations using magnetization variables

“…One noteworthy technicality is that the system (9) may not conserve momentum, so we must carefully control the zeroth Fourier coefficient in the case of periodic boundary conditions. This can be dealt with following the example of the diffusive Burgers equations [21], i.e. by justifying an a priori estimate of the form…”

Global well-posedness and regularity of 3D Burgers equation with multiplicative noise

“…For the multidimensional Burgers equations, Kiselev and Ladyzhenskaya [34] prove the existence and uniqueness of solution in the class of functions L ∞ (0, T ; L ∞ (O)) ∩ L 2 (0, T ; H 1 0 (O)). Inspired by [34], Pooley, Robinson [36] prove the global well-posedness for 3D Burgers equations in H 1 2 . When the viscosity tends to zero and the initial condition is zero, Bui [11] prove the convergence of solutions to the inviscid Burgers equations on a small time interval.…”