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In this paper we study classification of homogeneous solutions to the stationary Euler equation with locally finite energy. Written in the form $\bu = \n^\perp \Psi$, $\Psi(r,\th) = r^{\l} \psi(\th)$, for $\l >0$, we show that only trivial solutions exist in the range $0<\l<1/2$, i.e. parallel shear and rotational flows. In other cases many new solutions are exhibited that have hyperbolic, parabolic and elliptic structure of streamlines. In particular, for $\l>9/2$ the number of different non-trivial elliptic solutions is equal to the cardinality of the set $(2,\sqrt{2\l}) \cap \N$. The case $\l = 2/3$ is relevant to Onsager's conjecture. We underline the reasons why no anomalous dissipation of energy occurs for such solutions despite their critical Besov regularity $1/3$.

In this article we provide a proof of the Iitaka Cnm conjecture for algebraic fiber spaces over tori.

In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations.

We consider the problem of the random fluctuations in the solutions to elliptic PDEs with highly oscillatory random coefficients. In our setting, as the correlation length of the fluctuations tends to zero, the heterogeneous solution converges to a deterministic solution obtained by averaging. When the Green’s function to the unperturbed operator is sufficiently singular (i.e., not square integrable locally), the leading corrector to the averaged solution may be either deterministic or random, or both in a sense we shall explain. Our main application is the solution of an elliptic problem with random Robin boundary condition that may be used to model diffusion of signaling molecules through a layer of cells into a bulk of extracellular medium. The problem is then described by an elliptic pseudo-differential operator (a Dirichlet-to-Neumann operator) on the boundary of the domain with random potential. In the physical setting of a three dimensional extracellular medium on top of a two-dimensional surface of cells forming a layer of epithelium, we show that the approximate corrector to averaging consists of a deterministic correction plus a Gaussian field of amplitude proportional to the correlation length of the random medium. The result is obtained under some assumptions on the four-point correlation function in the medium. We provide examples of such random media based on Gaussian and Poisson statistics.