Mathematics

The theory of Poisson Vertex Algebras (PVAs) is a good framework to treat
Hamiltonian partial differential equations. A PVA consist of a pair
$(\mathcal{A},\{\cdot_{\lambda}\cdot\})$ of a differential algebra
$\mathcal{A}$ and a bilinear operation called the $\lambda$-bracket. We extend
the definition to the class of algebras $\mathcal{A}$ endowed with $d\geq 1$
commuting derivations. We call this structure a multidimensional PVA: it is a
suitable setting to the study of deformations of the Poisson bracket of
hydrodynamic type associated to the Euler's equation of motion of
$d$-dimensional incompressible fluids. We prove that for $d=2$ all the first
order deformations of such class of Poisson brackets are trivial.