A Lower Bound on Blowup Rates for the 3D Incompressible Euler Equation and a Single Exponential Beale-Kato-Majda Type Estimate
Abstract
We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in $${H^{s}(\mathbb {R}^3)}$$ , for $${s>\frac{5}{2}}$$ . Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on $${\|u(t)\|_{H^s}}$$ involving the length parameter introduced by Constantin in (SIAM Rev. 36(1):73-98, 1994). In particular, we derive lower bounds on the blowup rate of such solutions.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- August 2012
- DOI:
- 10.1007/s00220-012-1523-y
- arXiv:
- arXiv:1107.0435
- Bibcode:
- 2012CMaPh.314..265C
- Keywords:
-
- Vorticity;
- Euler Equation;
- Singular Integral Operator;
- Blowup Rate;
- Single Exponential;
- Mathematics - Analysis of PDEs;
- Mathematical Physics;
- 76B03
- E-Print:
- AMS Latex, 15 pages