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Bertrand Banos

Publications and source records attributed to Bertrand Banos.

8 recordsLinked to original sources
arXiv2015

Monge-Ampère Structures and the Geometry of Incompressible Flows

We show how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to a Monge-Ampère structure, and Burgers'-type vortices are a canonical class of solutions associated with this structure. The mapping of such solutions, which are characterised by a linear dependence of the third component of the velocity on the coordinate defining the axis of rotation, to solutions of the incompressible equations in two dimensions is also shown to be an example of a symmetry reduction The Monge-Ampère structure for incompressible flow in two dimensions is shown to be hypersymplectic.

math-ph↗
arXiv2011

Complex solutions of Monge-Ampère equations

We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equations.

math.DG↗
arXiv2006

Integrable geometries and Monge-Ampere equations

In this lecture delivered at the Integrable and Quantum Field Theory at Peyresq sixth meeting, we review the Lychagin's Monge-Ampere operators theory and exhibit the link it establishes between the classical problem of local equivalence for non linear partial differential equations and the problem of integrability of some geometrical structures.

math.DG↗
arXiv2005

K"ahler Geometry and the Navier-Stokes Equations

We study the Navier-Stokes and Euler equations of incompressible hydrodynamics in two and three spatial dimensions and show how the constraint of incompressiblility leads to equations of Monge--Ampère type for the stream function, when the Laplacian of the pressure is known. In two dimensions a Kähler geometry is described, which is associated with the Monge--Ampère problem. This Kähler structure is then generalised to `two-and-a-half dimensional' flows, of which Burgers' vortex is one example. In three dimensions, we show how a generalized Calabi--Yau structure emerges in a special case.

nlin.SI↗
arXiv2002

Nondegenerate Monge-Ampere structures in dimension 6

We define a nondegenerate Monge-Ampère structure on a 6-dimensional manifold as a pair $(Ω,ω)$, such that $Ω$ is a symplectic form and $ω$ is a 3-differential form which satisfies $ω\wedgeΩ=0$ and which is nondegenerate in the sense of Hitchin. We associate with such a pair a generalized almost (pseudo) Calabi-Yau structure and we study its integrability from the point of view of Monge-Ampère operators theory. The result we prove appears as an analogue of Lychagin and Roubtsov theorem on integrability of the almost complex or almost product structure associated with an elliptic or hyperbolic Monge-Ampère equation in the dimension 4. We study from this point of view the example of the Stenzel metric on the cotangent bundle of the sphere $S^3$.

math.DG↗
arXiv2002

Structures de Monge-Ampere symplectiques non degenerees en dimension 6

We define a non-degenerated Monge-Ampere structure on a 6-manifold associated with a Monge-Ampere equation as a couple (Ω,ω), such that Ωis a symplectic form and ωis a 3-differential form which satisfies ω\wedgeΩ=0 and which is non-degenerated in the sense of Hitchin. We associate with such a couple an almost (pseudo) Calabi-Yau structure and we study its integrability from the point of view of Monge-Ampere operators theory. The result we prove appears as an analogue of Lychagin and Roubtsov theorem on integrability of the almost complex or almost product structure associated with an elliptic or hyperbolic Monge-Ampere equation in the dimension 4. We study from this point of view the example of the Stenzel metric on T*S^3.

math.DG↗