We present results concerning the local existence, regularity and possible blow up of solutions to incompressible Euler and Navier-Stokes equations.

The following deals with the $T(b)$ theorems of David, Journé, and Semmes considered in a dyadic setting. We find sharp growth estimates for a global and a local dyadic $T(b)$ Theorem. We use multiscale analysis and Haar wavelets in the local case.

If $\mu$ is a finite complex measure in the complex plane $\mathbb{C}$ we denote by $C^\mu$ its Cauchy integral defined in the sense of principal value. The measure $\mu$ is called reflectionless if it is continuous (has no atoms) and $C^\mu=0$ at $\mu$-almost every point. We show that if $\mu$ is reflectionless and its Cauchy maximal function $C^\mu_*$ is summable with respect to $|\mu|$ then $\mu$ is trivial. An example of a reflectionless measure whose maximal function belongs to the ``weak" $L^1$ is also constructed, proving that the above result is sharp in its scale. We also give a partial geometric description of the set of reflectionless measures on the line and discuss connections of our results with the notion of sets of finite perimeter in the sense of De Giorgi.

The classical Tits construction provides models of the exceptionalsimple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group $S_4$ on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two different types of structurable algebras. Some models of exceptional Lie superalgebras are discussed too.

We study Sobolev embeddings in the Sobolev space $W^{1,p(\cdot)}(\Omega)$ with variable exponent satisfying $1\leqslant p(x) \leqslant n$. Since the exponent is allowed to reach the values $1$ and $n$, we need to introduce new techniques, combining weak- and strong-type estimates, and a new variable exponent target space scale which features a space of exponential type integrability instead of $L^\infty$ at the upper end.

This paper deals with the universal catenarity of a pullback construction ring. It seeks necessary and sufficient conditions for such a pullback to have the universal catenarity, improving some known results. Its main result leads to new examples of universally catenarian domains.

In the present paper we study the behaviour as $p$ goes to$1$ of the weak solutions to the problems

$$ \begin{cases}-\operatorname{div} \bigl(|\nabla u_p|^{p-2}\nabla u_p\bigr)=f \text{in } \Omega\\ u_p=0 \text{on } \partial\Omega, \end{cases}$$

where $\Omega$ is a bounded open set of ${\mathbb R}^N$ $(N\ge 2)$ with Lipschitz boundary and $p>1$. As far as the datum $f$ is concerned, we analyze several cases: the most general one is $f\in W^{-1,\infty}(\Omega)$. We also illustrate our results by means of remarks and examples.

A nonlinear degenerate Fokker-Planck equation in the whole space is analyzed. The existence of solutions to the corresponding implicit Euler scheme is proved, and it is shown that the semi-discrete solution converges to a solution of the continuous problem. Furthermore, the discrete entropy decays monotonically in time and the solution to the continuous problem is unique. The nonlinearity is assumed to be of porous-medium type. For the (given) potential, either a less than quadratic growth condition at infinity is supposed or the initial datum is assumed to be compactly supported. The existence proof is based on regularization and maximum principle arguments. Upper bounds for the tail behavior in space at infinity are also derived in the at-most-quadratic growth case.

In this article we use the twistor theory in order to build ``non standard'' complex structures (with a meaning which we define) on the products of $4$-manifolds with the sphere of dimension two. To that end, we enumerate the set of complex surfaces whose twistor space is $\mathcal C^\infty$-trivial. Among these surface we will study those which admit an anti-self-dual riemannian metric.

We show that an injective continuous map between planar regions which distorts vertices of equilateral triangles by a small amount is quasiconformal.

We relate some properties of complexifications of real analytic foliations with problems such that existence of first integrals or convergent normalizations. Holomorphic diffeomorphisms having an invariant real foliation play a crucial role.