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Project B5 • Biharmonic wave maps

Principal investigators

  Prof. Dr. Tobias Lamm  
  Prof. Dr. Birgit Schörkhuber  
  Prof. Dr. Roland Schnaubelt (7/2015 - 6/2019)

Project summary

In this project we study a biharmonic variant of the wave map system, more precisely the Euler-Lagrange equation of action functionals of the form \begin{equation} \label{action-functional} B(u)=\frac{1}{2}\int_0^T \int_M (|\partial_t u|^2-|\Delta_g u|^2)dv_gdt, \end{equation} with higher order potential energy. Here \(u:[0, T)\times M \to N\) is a map between Riemannian manifolds \( (M,g) \) and \( (N,h) \), where \(N \hookrightarrow \mathbb{R}^L\) is isometrically embedded in euclidean space.

Starting with \( M = \mathbb{R}^m \), we aim to obtain local wellposedness and global existence results for the biharmonic wave map system under suitable regularity and energy assumptions on the initial maps.

Introduction

Wave maps are maps \( u:[0,T)\times M \to N \) between Riemannian manifolds \( (M,g) \) and \( (N,h) \) which are critical points of the first-order action functional \[ W(u)=\frac{1}{2}\int_0^T\int_M (|\partial_t u|^2 - |\nabla u|^2)dv_g dt, \] where \(dv_g\) is the induced area element. They arise in various fields of mathematics, for instance in the description of the free motion of an elastic surface \( (M^{(2)},g) \) immersed into a fixed target manifold $ and in the mathematical theory of relativity. More precisely its relevance in physics include

  • the dimension \( d \geq 3 \), where they serve as a toy model for singularity formation of Einstein's equations,
  • the dimension \( d = 2 \), where they are part of a special case of \((3+1)\)-Einstein equations in hyperbolic coordinates (Einstein-wave map system).

The wave map system has been intensively studied in the past two decades as a model problem for nonlinear dispersion and singularity formation.

The model in our case generalizes free higher order wave equations as they appear in the linear part of models for the motion of elastic beams or plates. Similar as in the case of these equations (where \(M = \mathbb{R}^m\) and \(N = \mathbb{R}\)), the gradient term is replaced by curvature-like terms in the potential energy to model the movement of a rather thin, stiff and elastic object within the target manifold \(N\).

As a canonical first approximation, we use the Laplace–Beltrami operator \(\Delta_g\) on \((M,g)\) and obtain the action functional \eqref{action-functional}. The Euler-Lagrange equation of \eqref{action-functional} is then of beam-type, i.e. \( u : [0, T) \times M \to N \) satisfies \begin{align} \label{EL} \partial_t^2 u + \Delta^2_g u = f(u,\partial_tu,\nabla u,\nabla^2 u,\nabla^3u), \end{align} for an explicitly given nonlinearity \(f\) that is homogeneous in the derivatives of \(u\).

 

Besides global existence and local wellposedness, we have the following goals:

  • As in the case of wave maps, it is expected that solutions blow-up in finite time for certain initial values. We thus study the possible blow-up behavior of the solutions.
  • The previous goal can be achived by reducing to the corotational (i.e. symmetry invariant) biharmonic wave map problem for the sphere \( N = \mathbb{S}^{L-1}\). Here, self-similar solutions are prototypes of solutions that are singular in finite time. Moreover, we intend to investigate the (asymptotic) stability of solutions in this setting.

Results

The following existence result is part of the work of S. Herr, T. Lamm, and R. Schnaubelt.

 

In case \(N = \mathbb{S}^{l-1} \) is a round sphere (induced metric tensor) and \( M = \mathbb{R}^m \), there exists a global weak solution \[ u : \mathbb{R}^m \times [0,\infty) \to \mathbb{S}^{l-1},\quad u - u_0 \in C^0(\mathbb{R}; H^{\nu}(\mathbb{R}^m)),\quad \nu \in [0,2) \] of \eqref{EL} on \( (0, \infty) \times \mathbb{R}^m \) for initial data \((u(0), u_t(0)) = (u_0,u_1) : \mathbb{R}^m \to T\mathbb{S}^{l-1} \) with bounded energy \(\mathcal{E}(u_0, u_1)<\infty \) and \( u_1 \in T_{u_0}\mathbb{S}^{l-1} \) a.e. in \( \mathbb{R}^m \). In particular \(u\) conserves the energy \( \mathcal{E}(u, u_t)_{|t} = \mathcal{E}(u_0, u_1) \) for \( t > 0 \) and, equivalent to solving \eqref{EL} weakly, \(u\) solves \[ \partial_t^2u + \Delta^2 u \perp T_{u}\mathbb{S}^{l-1}\qquad\text{ on } [0, \infty) \times \mathbb{R}^m, \] in the sense of distributions.

The result is obtained from a penalization argument that implies the existence of limits of global solutions for the penalized equation by energy estimates (independent of the parameter of penalization). Further the geometry of the sphere is important in the approximation argument, in order to pass to a limiting weak solution by the use of a conservation law.

The following are partial results in the work of T.Schmid (ongoing PhD-project).

Let \( (N,h) \) be compact and isometrically embedded into euclidean space \((\mathbb{R}^l,\delta)\). Then there exist global solutions for the biharmonic wave map equation in the energy subcritical dimensions \(n = 1,2\) (by the use of energy bounds that exclude blow up in finite time). In the energy critical dimension \(n=4\) there exists a solution of the parabolic regularization \[ \partial_t^2 u + \Delta^2u + \varepsilon \Delta^2 u_t \perp T_{u}N \] with an energy bound that is uniform in \(\varepsilon>0\). With this, we hope to approximate a global weak solution in the critical dimension \(n=4\) (for more general target manifolds).

Preprints

  1. Biharmonic wave maps: local wellposedness in high regularity,
    S. Herr, T. Lamm, T. Schmid, R. Schnaubelt, January 2019 – BibTeX
  2. Energy bounds for biharmonic wave maps in low dimensions,
    T. Schmid, December 2018 – BibTeX
Project-specific staff
Name Phone E-Mail
+49 721 608-41696 tobias lammZrx8∂kit edu
   
+49 721 608-43697 tobias schmidVeo5∂kit edu
+49 721 608-48955 roland schnaubeltKll8∂kit edu
+49 721 608-46197 birgit schoerkhuberUji2∂kit edu