In this article we investigate Gevrey regularity of formal power series solutions for a certain class of nonlinear moment partial differential equations, the inhomogeneity of which is of a fixed Gevrey order with respect to the time variable. The results are a generalization of similar ones obtained for standard nonlinear partial differential equations as well as linear moment differential equations.
In this paper, we establish the global well-posedness of a strong solution to the three-dimensional Tropical climate model with fractional dissipation and damping. We prove the global existence and uniqueness of the strong solution under suitable assumptions on the dissipation and damping terms of the barotropic velocity. Our result is based on a priori energy estimates and interpolation inequalities. A key feature of this work is to minimize the required damping exponent when the dissipation is taken to be weaker. In addition, we note that the proof of uniqueness in this framework can also be applied to the Navier-Stokes and MHD equations.
Discretely self-similar solutions to Oberbeck-Boussinesq system with Newtonian gravitational field for large discretely self-similar initial data are constructed in this note, extending the construction of Brandolese and Karch on self-similar solutions. It follows the approach of Bradshaw and Tsai and finds an explicit a priori bound for the deviation from suitably revised profiles in similarity variables.
We define a hypergeometric series in m variables with p+(p-1)m parameters, which reduces to the generalized hypergeometric series _pF_p-1 when m=1, and to Lauricella's hypergeometric series F_C in m variables when p=2. We give a system of hypergeometric differential equations annihilating the series. Under some non-integral conditions on parameters, we give an Euler type integral representation of the series, and linearly independent p^m solutions to this system around a point near to the origin. We show that this system is of rank p^m, and determine its singular locus.
We introduce a configuration of a $q$-difference equation and characterize the variants of the $q$-hypergeometric equation, which were defined by Hatano-Matsunawa-Sato-Takemura, by configurations. We show integral solutions and series solutions for the variants of the $q$-hypergeometric equation.
