Quasi-periodic solutions of the 2D Euler equation
نویسندگان
چکیده
We consider the two-dimensional Euler equation with periodic boundary conditions. We construct time quasi-periodic solutions of this equation made of localized travelling profiles with compact support propagating over a stationary state depending on only one variable. The direction of propagation is orthogonal to this variable, and the support is concentrated on flat strips of the stationary state. The frequencies of the solution are given by the locally constant velocities associated with the stationary state.
منابع مشابه
A Recurrence Theorem on the Solutions to the 2d Euler Equation
In finite dimensions, the Poincaré recurrence theorem can be proved from the basic properties of a finite measure. In infinite dimensions, it is difficult to establish a natural finite measure, especially by extending a finite dimensional finite measure. A natural alternative is the Banach norm which can be viewed as a counterpart of the probability density. An interesting problem is to study t...
متن کاملChaotic response of the 2D semi-geostrophic and 3D quasi-geostrophic equations to gentle periodic forcing
Symmetries and Hamiltonian structure are combined with Melnikov’s method to show a set of exact solutions to the 2D semi-geostrophic equations in an elliptical tank respond chaotically to gentle periodic forcing of the domain eccentricity (or of the potential vorticity, for that matter) which are sinusoidal in time with nearly any period. A similar approach confirms the chaotic response of the ...
متن کاملInviscid Models Generalizing the 2d Euler and the Surface Quasi-geostrophic Equations
Any classical solution of the 2D incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component uj of the velocity field u is determined by the scalar θ th...
متن کاملAn incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations
We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time ...
متن کاملTime-Periodic Linearized Solutions of the Compressible Euler Equations and a Problem of Small Divisors
It has been unknown since the time of Euler whether or not time-periodic sound wave propagation is physically possible in the compressible Euler equations, due mainly to the ubiquitous formation of shock waves. The existence of such waves would confirm the possibility of dissipation free long distance signaling. Following our work in [27], we derive exact linearized solutions that exhibit the s...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Asymptotic Analysis
دوره 81 شماره
صفحات -
تاریخ انتشار 2013