A study of energy concentration and drain in incompressible fluids

In this paper we examine two opposite scenarios of energy behavior for solutions of the Euler equation. We show that if $u$ is a regular solution on a time interval $[0,T)$ and if $u \in L^rL^\infty$ for some $r\geq \frac{2}{N}+1$, where $N$ is the dimension of the fluid, then the energy at the time $T$ cannot concentrate on a set of Hausdorff dimension samller than $N - \frac{2}{r-1}$. The same holds for solutions of the three-dimensional Navier-Stokes equation in the range $5/3<r<7/4$. Oppositely, if the energy vanishes on a subregion of a fluid domain, it must vanish faster than (T-t)^{1-\d}, for any \d>0. The results are applied to find new exclusions of locally self-similar blow-up in cases not covered previously in the literature.Comment: an update of the previous versio

On formation of a locally self-similar collapse in the incompressible Euler equations

The paper addresses the question of existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the $L^p$-condition for velocity or vorticity and for a range of scaling exponents. In particular, in $N$ dimensions if in self-similar variables $u \in L^p$ and u \sim \frac{1}{t^{\a/(1+\a)}}, then the blow-up does not occur provided \a >N/2 or -1<\a\leq N/p. This includes the $L^3$ case natural for the Navier-Stokes equations. For \a = N/2 we exclude profiles with an asymptotic power bounds of the form |y|^{-N-1+\d} \lesssim |u(y)| \lesssim |y|^{1-\d}. Homogeneous near infinity solutions are eliminated as well except when homogeneity is scaling invariant.Comment: A revised version with improved notation, proofs, etc. 19 page

On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations

We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is well-posed. In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.Comment: We have modified the definition of Lipschitz well-posedness, in order to allow for a possible loss in regularity of the solution ma

On the second iterate for active scalar equations

We consider an iterative resolution scheme for a broad class of active scalar equations with a fractional power \gamma of the Laplacian and focus our attention on the second iterate. The main objective of our work is to analyze boundedness properties of the resulting bilinear operator, especially in the super-critical regime. Our results are two-fold: we prove continuity of the bilinear operator in BMO^{1-2\gamma} - a fractional analogue of the Koch-Tataru space; for equations with an even symbol we show that the B^{-\gamma}_{\infty,q} -regularity, where q > 2, is in a sense a minimal necessary requirement on the solution.Comment: This paper has been withdrawn by the author due to a crucial error in Section 3.