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The $C^{\a}$ regularity of a class hypoelliptic ultraparabolic equations  [PDF]
Wendong Wang,Liqun Zhang
Mathematics , 2008, DOI: 10.1007/s11425-009-0158-8
Abstract: We obtained the $C^{\a}$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form ${\ptl_t u}= \sum_{i,j=1}^{m_0}X_i(a_{ij}(x,t)X_j u)+X_0 u$. The result is proved by simplifying and generalizing our earlier arguments for the $C^{\a}$ regularity of homogeneous ultraparabolic equations.
The $C^{\a}$ regularity of a class of ultraparabolic equations  [PDF]
Liqun Zhang
Mathematics , 2005,
Abstract: We prove the $C^{\alpha}$ regularity for weak solutions to a class of ultraparabolic equation, with measurable coefficients. The results generalized our recent $C^{\alpha}$ regularity results of Prandtl's system to high dimensional cases.
The H?lder continuity of a class of 3-dimension ultraparabolic equations  [PDF]
Wang Wendong,Zhang Liqun
Mathematics , 2008,
Abstract: We obtained the $C^{\a}$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form $${\ptl_t u}= \ptl_x(a(x,y,t)\ptl_x u)+b_0(x,y,t)\ptl_x u+b(x,y,t)\ptl_y u,$$ which generalized our recent results on KFP equations.
A matrix differential Harnack estimate for a class of ultraparabolic equations  [PDF]
Hong Huang
Mathematics , 2013,
Abstract: Let $u$ be a positive solution of the ultraparabolic equation \begin{equation*} \partial_t u=\sum_{i=1}^n \partial_{x_i}^2 u+\sum_{i=1}^k x_i\partial_{x_{n+i}}u \hspace{8mm} \mbox{on} \hspace{4mm} \mathbb{R}^{n+k}\times (0,T), \end{equation*} where $1\leq k\leq n$ and $0
Gevrey Regularity for Solution of the Spatially Homogeneous Landau Equation  [PDF]
Hua Chen,Weixi Li,Chao-Jiang Xu
Mathematics , 2009,
Abstract: In this paper, we study the Gevrey class regularity for solutions of the spatially homogeneous Landau equations in the hard potential case and the Maxwellian molecules case.
Degree of Regularity of Linear Homogeneous Equations  [PDF]
Kavish Gandhi,Noah Golowich,László Miklós Lovász
Mathematics , 2013, DOI: 10.4310/JOC.2014.v5.n2.a5
Abstract: We define a linear homogeneous equation to be strongly r-regular if, when a finite number of inequalities is added to the equation, the system of the equation and inequalities is still r-regular. In this paper, we show that, if a linear homogeneous equation is r-regular, then it is strongly r-regular. In 2009, Alexeev and Tsimerman introduced a family of equations, each of which is (n-1)-regular but not n-regular, verifying a conjecture of Rado from 1933. These equations are actually strongly (n-1)-regular as an immediate corollary of our results.
Global existence, regularity and a probabilistic scheme for a class of ultraparabolic Cauchy problems  [PDF]
Christian Fries,Joerg Kampen
Mathematics , 2010,
Abstract: In this paper we establish a constructive method in order to show global existence and regularity for a class of degenerate parabolic Cauchy problems which satisfy a weak Hoermander condition on a subset of the domain where the data are measurable and which have regular data on the complementary set of the domain. This result has practical incentives related to the computation of Greeks in reduced LIBOR market models, which are standard computable approximations of the HJM-description of interest rate markets. The method leads to a probabilistic scheme for the computation of the value function and its sensitivities based on Malliavin calculus. From a practical perspective the main contribution of the paper is an Monte-Carlo algorithm which includes weight corrections for paths which move in time into a region where a (weak) Hoermander condition holds.
On the analyticity and Gevrey class regularity up to the boundary for the Euler Equations  [PDF]
Igor Kukavica,Vlad Vicol
Mathematics , 2010, DOI: 10.1088/0951-7715/24/3/004
Abstract: We consider the Euler equations in a three-dimensional Gevrey-class bounded domain. Using Lagrangian coordinates we obtain the Gevrey-class persistence of the solution, up to the boundary, with an explicit estimate on the rate of decay of the Gevrey-class regularity radius.
Gevrey regularity for a class of dissipative equations with analytic nonlinearity  [PDF]
Hantaek Bae,Animikh Biswas
Mathematics , 2014,
Abstract: In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations, and allowing the dissipation operator to be a fractional Laplacian. Moreover, we derive global (in time) estimates in Gevrey norms which yields decay of higher order derivatives which are optimal. Applications include (temporal) decay of solutions in higher Sobolev norms for a large class of equations including the Navier-Stokes equations, the subcritical quasi-geostrophic equations, nonlinear heat equations with fractional dissipation, a variant of the Burgers' equation with a cubic or higher order nonlinearity, and the generalized Cahn-Hilliard equation. The decay results for the last three cases seem to be new while our approach provides an alternate proof for the recently obtained $L^p\, (1
Propagation of Gevrey Regularity for a Class of Hypoelliptic Equations  [PDF]
Antonio Bove,David S. Tartakoff
Mathematics , 1994,
Abstract: We prove results on the propagation of Gevrey and analytic wave front sets for a class of $C^\infty$ hypoelliptic equations with double characteristics.
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