arXiv research

Study geometric singular solutions of generalized Monge-Ampère equations.

In this paper, a type of integrable evolution equation--the generalized Landau-Lifshitz equation into $S^n$ is considered. We deal with this equation from a geometric point of view by rewriting it in a geometric form. Through the geometric energy method, we show the global well-posedness of the corresponding Cauchy pro…

The article derives gradient estimations for semilinear equations on geometric flows.

Modified Gibbs-Helmholtz equation geometric models for thermodynamics.

Geometrically describes Jacobi equations for field theories with dissipation.

Geometric mechanics approach to constrained and floating multibody systems using Hamel's equations.

Extends gradient estimates for heat equation under Finsler geometric flows.

Geometric flow on curves in S^3 generates YO equations solutions.

We write the Dirac equation in curved 4-dimensional Lorentzian spacetime using concepts from the analysis of partial differential equations as opposed to geometric concepts.

The paper explains how microlocal analysis solves geometric inverse problems.

We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein met…

In this article we develop geometric versions of the classical Langevin equation on regular submanifolds in euclidean space in an easy, natural way and combine them with a bunch of applications. The equations are formulated as Stratonovich stochastic differential equations on manifolds. The first version of the geometr…

New framework tackles geometric structure existence and classification.

A complete classification of isotropic vector equations of the geometric type that possess higher symmetries is proposed. New examples of integrable multi-component systems of the geometric type and their auto-Backlund transformations are found.

New method finds precise late-time behavior of wave equations.

Extends Zeitlin's model to 3-D axisymmetric Euler equations.

Introduces geometric control theory for students.

Study geometric equations on cohomogeneity one manifolds near singular orbits.

The paper estimates gradients for a weighted parabolic equation under geometric flow.

Notes on relative algebroids for geometric problems.

Superintegrable systems on surfaces are classified geometrically.

In this paper we study the difference between algebraic and geometric solutions of the hyperbolic Dehn filling equations for ideally triangulated 3-manifolds. We show that any geometric solution is an algebraic one, and we prove the uniqueness of the geometric solutions. Then we do explicit calculations for three inter…

Geometrically reformulates GENERIC stochastic dynamics.

New theorems prove uniqueness of solutions to geometric PDEs.

Geometrical properties of holonomic and non holonomic varieties defined by the Pfaff equations connected with a first order systems of differential equations are studied. The Riemann extensions of affine connected spaces for investigation of geodesics and asymptotic lines are used.

We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.

Geometric tools study two- and threepeakons in Camassa-Holm equation.

Geometrically proves WKB solutions of Schrödinger equations are resurgent.

Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (eg. direction, shape) or temporal (eg. speed) features of trajectories rather than trajectory's representation as a whole. This work is about empirically supported mathematical ideas behind splitting and merging…

New approach for uniform estimates in complex equations.

Study geometric step options with jumps, deriving pricing equations and characterizations.

In the present paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation under general geometric flow on complete noncompact manifolds.

This research smooths out fluid equations to avoid sudden shocks.

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

Reinterprets Schrödinger equation using Cartan connection for geometric investigation.

Solves geometric problems using fully nonlinear equations and Morse theory.

The 1-d Schrodinger flow on 2-sphere, the Gauss-Codazzi equation for flat Lagrangian submanifolds in C^n, and the space-time monopole equation are all examples of geometric soliton equations. The linear systems with a spectral parameter (Lax pair) associated to these equations satisfy the reality condition associated t…

Geometric structures help in understanding thermodynamics.

We study a class of fourth order geometric equations defined on a 4-dimensional compact Riemannian manifold which includes the Q-curvature equation. We obtain sharp estimates on the difference near the blow-up points between a bubbling sequence of solutions and the standard bubble.

Survey new geometric inequalities for the Monge-Ampere equation.

Survey on geometric, analytic, and topological aspects of 4D equations.

Geometric focusing affects dispersive estimates for Schrödinger and wave equations.

Geometrically interprets two equations, showing their equivalence and providing solutions.

New geometric flow equations describe how space-time dimensions change.

The paper provides gradient estimates for a parabolic equation under Finsler geometric flows.

Solves initial boundary value problem for vacuum Einstein equations and proves geometric uniqueness.

New geometric transformations link discrete and continuous curve motions.

The commuting vector fields approach, devised for strichartz estimates in [13], was developed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in ${\mathbb R}^{1+3}$ by Klainerman and Rodnianski. Via this approach they obtained the l…