Finite Dimensional Uniform Attractors for the Nonautonomous Camassa-Holm Equations

Abstract

We consider the uniform attractors for the three-dimensional nonautonomous Camassa-Holm equations in the periodic box $\Omega ={\left[0,L\right]}^3$. Assuming $f=f\left(x,t\right)\in L_{\mathrm{loc}}^2\left(\left(0,T\right);D\left(A^{-1/2}\right)\right)$, we establish the existence of the uniform attractors in $D\left(A^{1/2}\right)$ and $D\left(A\right)$. The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.