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Register a new userExistence and regularity of co-rotating and travelling global solutions for the generalized SQG equation
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By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and travelling global solutions for the gSQG equation, which extends the result of Hmidi and Mateu cite{HM} to $alphain[1,2)$. Moreover, we prove the $C^infty$ regularity of vortices boundary, and show the convexity of each vortices component.
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In this paper, we study the existence of rotating and traveling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation. The solutions are obtained by maximization of the energy over the set of rearrangements of a fixed function. The rotating solutions take the form of co-rotating vortices with $N$-fold symmetry. The traveling-wave solutions take the form of translating vortex pairs. Moreover, these solutions constitute the desingularization of co-rotating $N$ point vortices and counter-rotating pairs. Some other quantitative properties are also established.
In this paper, we study the existence of global classical solutions to the generalized surface quasi-geostrophic equation. By using the variational method, we provide some new families of global classical solutions for to the generalized surface quasi-geostrophic equation. These solutions mainly consist of rotating solutions and travelling-wave solutions.
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}.
This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. We also consider a non-degenerate modification of the endpoint case in which the velocity is less smooth than the advected scalar by slightly more than one order. The existence and uniqueness theory of these equations in the borderline Sobolev spaces is addressed, as well as the instantaneous smoothing effect of their corresponding solutions. In particular, it is shown that solutions emanating from initial data belonging to these Sobolev classes immediately enter a Gevrey class. Such results appear to be the first of its kind for a quasilinear parabolic equation whose coefficients are of higher order than its linear term; they rely on an approximation scheme which modifies the flux in such a way that preserves the underlying commutator structure lost by having to work in the critical space setting, as well as delicate adaptations of well-known commutator estimates to Gevrey classes.
By applying implicit function theorem on contour dynamics, we prove the existence of co-rotating and travelling patch solutions for both Euler and the generalized surface quasi-geostrophic equation. The solutions obtained constitute a desingularization of points vortices when the size of patch support vanishes. In particular, solutions constructed in this paper consist of doubly connected components, which is essentially different from all known results.
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