Project Details
Abstract
(1) Wider research context / theoretical framework
Biwave maps are the analogue of biharmonic maps in the case of a domain manifold with Lorentzian metric. They can also be thought of as a fourth order generalization of wave maps which are a second order hyperbolic equation. Such kind of maps appear in quantum field theory and elasticity but are also interesting from the pure mathematics perspective as they are given by a semilinear hyperbolic partial differential equation of fourth order in both time and spatial variables.
So far biwave maps did not receive much attention and this project will initiate a first deep mathematical investigation of the latter. We will show up many fascinating properties of biwave maps and point out unexpected differences to both wave and biharmonic maps.
(2) Hypotheses / research questions / objectives
In order to obtain a basic intuition on the biwave map equation we will first address the following aspects:
a)Conserved energies
b)Local well-posedness for small initial data
c)Finding explicit solutions
As every wave map is also a solution of the biwave map equation we will find conditions that force biwave maps to be wave maps and compare them to the conditions that force biharmonic maps to be harmonic.
In the core part of the project we will extend several existence results for wave maps and biharmonic maps to biwave maps. We will establish the existence of biwave maps
a)from Minkowski space to spheres
b)from Minkowski space to complete Riemannian manifolds
c)in the equivariant setup
d)on expanding spacetimes
We will also perform a first singularity analysis of biwave maps and expect to find interesting differences compared to the singularity formation of wave maps.
The climax of this project will be the investigation of an action functional that interpolates between wave and biwave maps.
(3) Approach / methods
In order to address the hyperbolic nature of the biwave map equation we will utilize tools from the analysis of wave maps and combine them with geometric methods from the study of biharmonic maps which are welladapted to fourth order equations.
To answer the analytic questions of the project we will make use of suitable energy estimates. The geometric methods include the use of suitable conformal transformations of the domain metric as well as the construction of warped product manifolds with a suitable warping function.
(4) Level of originality / innovation
As there has not been any rigorous mathematical investigation of the analytic aspects of the biwave map equation this project has the potential to reveal fascinating new directions of research. The results obtained in this project will also shed new light on both biharmonic and wave maps.
(5) Primary researchers involved
The principal investigator of this project will be Volker Branding.
Biwave maps are the analogue of biharmonic maps in the case of a domain manifold with Lorentzian metric. They can also be thought of as a fourth order generalization of wave maps which are a second order hyperbolic equation. Such kind of maps appear in quantum field theory and elasticity but are also interesting from the pure mathematics perspective as they are given by a semilinear hyperbolic partial differential equation of fourth order in both time and spatial variables.
So far biwave maps did not receive much attention and this project will initiate a first deep mathematical investigation of the latter. We will show up many fascinating properties of biwave maps and point out unexpected differences to both wave and biharmonic maps.
(2) Hypotheses / research questions / objectives
In order to obtain a basic intuition on the biwave map equation we will first address the following aspects:
a)Conserved energies
b)Local well-posedness for small initial data
c)Finding explicit solutions
As every wave map is also a solution of the biwave map equation we will find conditions that force biwave maps to be wave maps and compare them to the conditions that force biharmonic maps to be harmonic.
In the core part of the project we will extend several existence results for wave maps and biharmonic maps to biwave maps. We will establish the existence of biwave maps
a)from Minkowski space to spheres
b)from Minkowski space to complete Riemannian manifolds
c)in the equivariant setup
d)on expanding spacetimes
We will also perform a first singularity analysis of biwave maps and expect to find interesting differences compared to the singularity formation of wave maps.
The climax of this project will be the investigation of an action functional that interpolates between wave and biwave maps.
(3) Approach / methods
In order to address the hyperbolic nature of the biwave map equation we will utilize tools from the analysis of wave maps and combine them with geometric methods from the study of biharmonic maps which are welladapted to fourth order equations.
To answer the analytic questions of the project we will make use of suitable energy estimates. The geometric methods include the use of suitable conformal transformations of the domain metric as well as the construction of warped product manifolds with a suitable warping function.
(4) Level of originality / innovation
As there has not been any rigorous mathematical investigation of the analytic aspects of the biwave map equation this project has the potential to reveal fascinating new directions of research. The results obtained in this project will also shed new light on both biharmonic and wave maps.
(5) Primary researchers involved
The principal investigator of this project will be Volker Branding.
Status | Finished |
---|---|
Effective start/end date | 13/01/21 → 12/01/25 |
Keywords
- wave maps
- biwave maps
- biharmonic maps
- global solutions
- blowup
- harmonic maps