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 so as to preserve 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.
In this article, we derived an equality for CR-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a CR-warped product submanifolds in Ka¨hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of CR-warped product submanifolds in the Ka¨hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor.
<p style='text-indent:20px;'>This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> is of lower singularity, i.e., <inline-formula><tex-math id="M3">\begin{document}$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a logarithmic smoothing operator and <inline-formula><tex-math id="M5">\begin{document}$ \beta \in [0, 1] $\end{document}</tex-math></inline-formula>. We complete this study by considering the more singular regime <inline-formula><tex-math id="M6">\begin{document}$ \beta\in(1, 2) $\end{document}</tex-math></inline-formula>. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.</p>
A generalized Chen-type inequality and corresponding equality consequences for sequential warped products are proved in this paper. These results, which are based on such warped products having factors holomorphic, totally real and pointwise slant, extend Chen-type inequality for various warped products on nearly Kähler manifolds. Moreover, we also examine the related special cases.
The main purpose of this paper is to study transversal hypersurface (briefly, $\mathcal{T}$-hypersurface) $P$ of a paraSasakian manifold $M$. We derive results allied with totally geodesic and totally umbilical $\mathcal{T}$-hypersurface of $M$. The necessary and sufficient condition for normality of $(\mathfrak{f},\mathfrak{g},\mu,\upsilon,\delta)$-structure is established. Examples of $\mathcal{T}$-hypersurface are also illustrated.
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