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Register a new userDimension estimates for the attractor of the regularized damped Euler equations on the sphere
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We prove existence of the global attractor of the damped and driven Euler--Bardina equations on the 2D sphere and on arbitrary domains on the sphere and give explicit estimates of its fractal dimension in terms of the physical parameters.
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We prove existence of the global attractor of the damped and driven 2D Euler--Bardina equations on the torus and give an explicit two-sided estimate of its dimension that is sharp as $alphato0^+$.
The dependence of the fractal dimension of global attractors for the damped 3D Euler--Bardina equations on the regularization parameter $alpha>0$ and Ekman damping coefficient $gamma>0$ is studied. We present explicit upper bounds for this dimension for the case of the whole space, periodic boundary conditions, and the case of bounded domain with Dirichlet boundary conditions. The sharpness of these estimates when $alphato0$ and $gammato0$ (which corresponds in the limit to the classical Euler equations) is demonstrated on the 3D Kolmogorov flows on a torus.
We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension when the forcing term belongs to the whole scale of homogeneous Sobolev spaces from -1 to 1
In this paper we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the un-damped set. We show that if replacing the damping term with a higher-order emph{complex absorbing potential} gives an operator enjoying polynomial resolvent bounds on the real axis, then the resolvent associated to our damped problem enjoys bounds of the same order. It is known that the necessary estimates with complex absorbing potential can also be obtained via gluing from estimates for corresponding non-compact models.
We study spectral instability of steady states to the linearized 2D Euler equations on the torus written in vorticity form via certain Birman-Schwinger type operators $K_{lambda}(mu)$ and their associated 2-modified perturbation determinants $mathcal D(lambda,mu)$. Our main result characterizes the existence of an unstable eigenvalue to the linearized vorticity operator $L_{rm vor}$ in terms of zeros of the 2-modified Fredholm determinant $mathcal D(lambda,0)=det_{2}(I-K_{lambda}(0))$ associated with the Hilbert Schmidt operator $K_{lambda}(mu)$ for $mu=0$. As a consequence, we are also able to provide an alternative proof to an instability theorem first proved by Zhiwu Lin which relates existence of an unstable eigenvalue for $L_{rm vor}$ to the number of negative eigenvalues of a limiting elliptic dispersion operator $A_{0}$.
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