A. Alexandrou Himonas

The Cauchy problem for the Novikov equation

This work studies the initial value problem for a Camassa–Holm type equation with cubic nonlinearities that has been recently discovered by Vladimir Novikov to be integrable. For s > 3/2, using a Galerkin-type approximation method, it is shown that this equation is well-posed in Sobolev spaces Hs on both the line and the circle with continuous dependence on initial data. Furthermore, it is proved that this dependence is optimal by showing that the data-to-solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions in conjunction with well-posedness estimates.

Non-Uniform Dependence for the Periodic CH Equation

We show that the solution map of the periodic CH equation is not uniformly continuous in Sobolev spaces with exponent greater than 3/2. This extends earlier results to the whole range of Sobolev exponents for which local well-posedness of CH is known. The crucial technical tools used in the proof of this result are a sharp commutator estimate and a multiplier estimate in Sobolev spaces of negative index.

Hölder continuity of the solution map for the Novikov equation

The Novikov equation (NE) has been discovered recently as a new integrable equation with cubic nonlinearities that is similar to the Camassa-Holm and Degasperis-Procesi equations, which have quadratic nonlinearities. NE is well-posed in Sobolev spaces Hs on both the line and the circle for s > 3/2, in the sense of Hadamard, and its data-to-solution map is continuous but not uniformly continuous. This work studies the continuity properties of NE further. For initial data in Hs, s > 3/2, it is shown that the solution map for NE is Holder continuous in Hr-topology for all 0 ⩽ r < s with exponent α depending on s and r.

Non-analytic hypoellipticity in the presence of symplecticity

Here we construct non-analytic solutions to a class of hypoelliptic operators with symplectic characteristic set and in the form of a sum of squares of real analytic vector fields.

GLOBAL ANALYTIC REGULARITY FOR SUMS OF SQUARES OF VECTOR FIELDS

We consider a class of operators in the form of a sum of squares of vector fields with real analytic coefficients on the torus and we show that the zero order term may influence their global analytic hypoellipticity. Also we extend a result of Cordaro-Himonas.

The cauchy problem for a shallow water type equation

We prove existence and uniqueness of local and global solutions of the periodic Cauchy problem for a higher order shallow water type equation under low regularity initial data. Using Fourier analysis we first prove local estimates in appropriate spaces and then use a contraction mapping argument and a conserved norm to get global existence.

Hölder Continuity for the Fokas–Olver–Rosenau–Qiao Equation

It has been shown that the Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation is well-posed for initial data

On global hypoellipticity of degenerate elliptic operators

u_0\in H^s