On well-posedness of the Degasperis-Procesi equation
A. Alexandrou Himonas Curtis Holliman
Discrete & Continuous Dynamical Systems - A 2011, 31(2): 469-488 doi: 10.3934/dcds.2011.31.469
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
Discrete & Continuous Dynamical Systems - A 2007, 19(3): 515-529 doi: 10.3934/dcds.2007.19.515
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
Conference Publications 2011, 2011(Special): 634-642 doi: 10.3934/proc.2011.2011.634
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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