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Register a new userThe Surface Quasi-Geostrophic Equation with Random Diffusion
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Consider the surface quasi-geostrophic equation with random diffusion, white in time. We show global existence and uniqueness in high probability for the associated Cauchy problem satisfying a Gevrey type bound. This article is inspired by recent work of Glatt-Holtz and Vicol.
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We are concerned with the existence of periodic travelling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation(including incompressible Euler equation), known as von Karman vortex street. These solutions are of $C^1$ type, and are obtained by studying a semilinear problem on an infinite strip whose width equals to the period. By a variational characterization of solutions, we also show the relationship between vortex size, travelling speed and street structure. In particular, the vortices with positive and negative intensity have equal or unequal scaling size in our construction, which constitutes the regularization for Karman point vortex street.
For the generalized surface quasi-geostrophic equation $$left{ begin{aligned} & partial_t theta+ucdot abla theta=0, quad text{in } mathbb{R}^2 times (0,T), & u= abla^perp psi, quad psi = (-Delta)^{-s}theta quad text{in } mathbb{R}^2 times (0,T) , end{aligned} right. $$ $0<s<1$, we consider for $kge1$ the problem of finding a family of $k$-vortex solutions $theta_varepsilon(x,t)$ such that as $varepsilonto 0$ $$ theta_varepsilon(x,t) rightharpoonup sum_{j=1}^k m_jdelta(x-xi_j(t)) $$ for suitable trajectories for the vortices $x=xi_j(t)$. We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem $$(-Delta)^sW = (W-1)^gamma_+, quad text{in } mathbb{R}^2, quad 1<gamma < frac{1+s}{1-s}$$ whose existence and uniqueness have recently been proven in cite{chan_uniqueness_2020}.
We continue our study of the dynamics of a nearly inviscid periodic surface quasi-geostrophic equation. Here we consider a slightly diffusive stochastic SQG equation of the form begin{equation*} begin{cases} dtheta_t + |D|^{2delta}theta_t,dx + (u_t cdot abla)theta_t,dx + |D|^{delta}dW_t = 0 u_t = abla^perp|D|^{-1}theta_t. end{cases} end{equation*} We construct global energy solutions as introduced by P. Goncalves and M. Jara (2014) for any $delta > 0$, so that any small amount of diffusion permits us to construct solutions. We show moreover that pathwise uniqueness of these energy solutions holds in the presence of sufficiently high diffusion $delta > frac32$.
In this paper, we construct smooth travelling counter-rotating vortex pairs with circular supports for the generalized surface quasi-geostrophic equation. These vortex pairs are analogues of the Lamb dipoles for the two-dimensional incompressible Euler equation. The solutions are obtained by maximization of the energy over some appropriate classes of admissible functions. We establish the uniqueness of maximizers and compactness of maximizing sequences in our variational setting. Using these facts, we further prove the orbital stability of the circular vortex pairs for the gSQG equation.
We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to $infty$. Both results rely on the Strichartz-type estimates for the corresponding linear equation.
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