On well-posedness of the Degasperis-Procesi equation
A. Alexandrou Himonas Curtis Holliman
It is shown in both the periodic and the non-periodic cases that the data-to-solution map for the Degasperis-Procesi (DP) equation is not a uniformly continuous map on bounded subsets of Sobolev spaces with exponent greater than 3/2. This shows that continuous dependence on initial data of solutions to the DP equation is sharp. The proof is based on well-posedness results and approximate solutions. It also exploits the fact that DP solutions conserve a quantity which is equivalent to the $L^2$ norm. Finally, it provides an outline of the local well-posedness proof including the key estimates for the size of the solution and for the solution's lifespan that are needed in the proof of the main result.
keywords: non-uniform dependence on initial data conserved quantities. DP equation well-posedness approximate solutions Sobolev spaces commutator estimate Cauchy problem
On unique continuation for the modified Euler-Poisson equations
A. Alexandrou Himonas Gerard Misiołek Feride Tiǧlay
It is shown that if a classical solution $(u, n)$ of the modified Euler-Poisson equation (mEP) in one space dimension is such that $u$, $u_x$ and $n$ are initially decaying exponentially and for some later time the first component $u$ is also decaying exponentially, then $n$ must be identically equal to zero and $u$ must be a solution to the Burgers equation. In particular, if $n$ and $u$ are initially compactly supported then $n$ can not be compactly supported at any later time, unless $n$ is identically equal to zero and $u$ is a solution to the Burgers equation. It is also shown that the mEP equations are locally well-posed in $H^s \times H^{s-1}$ for $s>5/2$.
keywords: local well-posedness exponential decay unique continuation Burgers equation Camassa-Holm equation Modified Euler-Poisson equations Sobolev spaces.
Anisotropic Gevrey regularity for mKdV on the circle
Heather Hannah A. Alexandrou Himonas Gerson Petronilho
It is shown that the solution to the Cauchy problem for the modifi ed Korteweg-de Vries equation with initial data in an analytic Gevrey space $G^\sigma$, $\sigma \>= 1$, as a function of the spacial variable belongs to the same Gevrey space. However, considered as function of time the solution does not belong to $G^\sigma$. In fact, it belong to $G^(3\sigma)$ and not to any Gevrey space $G^r$, 1 $\<= r$ < 3$\sigma$.
keywords: Modi ed KdV equation Sobolev spaces periodic Gevrey regularity Cauchy problem mKdV

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