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  • on the dimension of the Singular Set of solutions to the navier stokes equations
    Communications in Mathematical Physics, 2012
    Co-Authors: James C. Robinson, Witold Sadowski
    Abstract:

    In this paper we prove that if a suitable weak solution u of the Navier–Stokes equations is an element of \({L^w(0,T;L^s(\mathbb{R}^3))}\), where 1 ≤ 2/w + 3/s ≤ 3/2 and 3 < w, s < ∞, then the box-counting dimension of the Set of space-time Singularities is no greater than max{w, s}(2/w + 3/s − 1). We also show that if \({\nabla u \in L^w(0,T;L^s(\Omega))}\) with 2 < s ≤ w < ∞, then the Hausdorff dimension of the Singular Set is bounded by w(2/w + 3/s − 2). In this way we link continuously the bounds on the dimension of the Singular Set that follow from the partial regularity theory of Caffarelli, Kohn, & Nirenberg (Commun. Pure Appl. Math. 35:771–831, 1982) to the regularity conditions of Serrin (Arch. Ration. Mech. Anal. 9:187–191, 1962) and Beirao da Veiga (Chin. Ann. Math. Ser. B 16(4):407–412, 1995).

  • On the Dimension of the Singular Set of Solutions to the Navier–Stokes Equations
    Communications in Mathematical Physics, 2011
    Co-Authors: James C. Robinson, Witold Sadowski
    Abstract:

    In this paper we prove that if a suitable weak solution u of the Navier–Stokes equations is an element of \({L^w(0,T;L^s(\mathbb{R}^3))}\), where 1 ≤ 2/w + 3/s ≤ 3/2 and 3 < w, s < ∞, then the box-counting dimension of the Set of space-time Singularities is no greater than max{w, s}(2/w + 3/s − 1). We also show that if \({\nabla u \in L^w(0,T;L^s(\Omega))}\) with 2 < s ≤ w < ∞, then the Hausdorff dimension of the Singular Set is bounded by w(2/w + 3/s − 2). In this way we link continuously the bounds on the dimension of the Singular Set that follow from the partial regularity theory of Caffarelli, Kohn, & Nirenberg (Commun. Pure Appl. Math. 35:771–831, 1982) to the regularity conditions of Serrin (Arch. Ration. Mech. Anal. 9:187–191, 1962) and Beirao da Veiga (Chin. Ann. Math. Ser. B 16(4):407–412, 1995).

  • decay of weak solutions and the Singular Set of the three dimensional navier stokes equations
    arXiv: Analysis of PDEs, 2006
    Co-Authors: James C. Robinson, Witold Sadowski
    Abstract:

    We consider the behaviour of weak solutions of the unforced three-dimensional Navier-Stokes equations, under the assumption that the initial condition has finite energy ($\|u\|^2=\int|u|^2$) but infinite enstrophy ($\|Du\|^2=\int|{\rm curl} u|^2$). We show that this has to be reflected in the solution for small times, so that in particular $\|Du(t)\|\to+\infty$ as $t\to0$. We also give some limitations on this `backwards blowup', and give an elementary proof that the upper box-counting dimension of the Set of Singular times can be no larger than one half. Although similar in flavour, this final result neither implies nor is implied by Scheffer's result that the 1/2-dimensional Hausdorff measure of the Singular Set is zero.

Jan Kristensen - One of the best experts on this subject based on the ideXlab platform.

Hi Jun Choe - One of the best experts on this subject based on the ideXlab platform.

John L. Lewis - One of the best experts on this subject based on the ideXlab platform.