Volume 85, Nº 4 (2021)

We describe the symmetry group of a multidimensional continued fraction. As a multidimensional generalization of continued fractions we consider Klein polyhedra. We distinguish two types of symmetries: Dirichlet symmetries, which correspond to the multiplication by units of the respective extension of $\mathbb{Q}$, and so-called palindromic symmetries. The main result is a criterion for a two-dimensional continued fraction to have palindromic symmetries, which is analogous to the well-known criterion for the continued fraction of a quadratic irrationality to have a symmetric period.

We consider the Cauchy problem for a model partial differential equation of third order with non-linearityof the form $|u|^q$, where $u=u(x,t)$ for $x\in\mathbb{R}^3$ and $t\ge 0$. We construct a fundamental solution for thelinear part of the equation and use it to obtain analogues of Green's third formula for elliptic operators, firstin a bounded domain and then in unbounded domains. We derive an integral equation for classical solutions of theCauchy problem. A separate study of this equation yields that it has a unique inextensible-in-time solutionin weighted spaces of bounded and continuous functions. We prove that every solution of the integral equation is a local-in-time weak solution of the Cauchy problem provided that $q>3$. When $q\in(1,3]$, we use Pokhozhaev'snon-linear capacity method to show that the Cauchy problem has no local-in-time weak solutions for a large class ofinitial functions. When $q\in(3,4]$, this method enables us to prove that the Cauchy problemhas no global-in-time weak solutions for a large class of initial functions.

We consider equations for the three-dimensional unsteady motion of mixtures of viscous compressible heat-conducting fluids in the multi-velocity approach. We prove theexistence, globally in time and the input data, of a generalized (dissipative) solutionof the initial-boundary value problem corresponding to flows in a bounded domain.

We answer a question of Filip and Tosatti concerning a basepoint-free theorem for transcendental$(1,1)$-classes on compact Kähler threefolds.