Project B5 • Geometric wave equations (7/2015 - 6/2023)

Principal investigators

 

  Prof. Dr. Tobias Lamm (7/2015 - 6/2023)
  Prof. Dr. Birgit Schörkhuber (7/2019 - 5/2021)
  Prof. Dr. Roland Schnaubelt (7/2015 - 6/2019)

Project summary

In this project we study geometric wave equations such as wave maps, biharmonic wave maps and also the Yang–Mills equation. These nonlinear equations arise naturally in various fields such as particle physics and general relativity. In the past decades second order geometric wave equations, in particular wave maps, have been the subject of intensive research and there exists a vast literature on various aspects related to well-posedness theory, blow-up phenomena, and long-time dynamics. One main goal of this project is to extend some of these groundbreaking results to the higher order case, in particular to biharmonic wave maps. Another main objective is to further advance the understanding of singularity formation for supercritical wave maps and the Yang–Mills equation, as well as of wave equations with power nonlinearities as their toy model counterparts. In order to obtain these results we combine techniques from harmonic analysis, spectral theory, and many other subfields of Analysis.

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 \begin{equation}\label{WM-functional} W(u)=\frac{1}{2}\int_0^T\int_M (|\partial_t u|^2 - |\nabla u|^2)dv_g dt, \end{equation} where $dv_g$ is the induced area element. They arise, for instance, in the description of the free motion of an elastic surface immersed into a fixed target manifold $N$ and also 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 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 $M=\mathbb{R}^n$ 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.

If one now replaces the gradient term in (\ref{WM-functional}) with the Laplace-Beltrami operator $\Delta_g$ on $(M,g)$ one obtains the action-functional of the so-called biharmonic wave maps. The Euler-Lagrange equation for these kind of maps is now of beam-type, i.e $u:[0,T)\times M \to N$ satisfies \begin{equation} \label{EL} \partial_t^2+\Delta_g^2 u = f(u,\partial_tu,\nabla u, \nabla^2u, \nabla^3 u) \end{equation} for an explicitly given nonlinearity $f$ that is homogeneous in the derivatives of $u$.

Goals

  • Prove (co-dimension) stable self-similar blow-up in energy supercritical nonlinear geometric wave equations (in particular, for the Yang–Mills equation, co-rotational wave maps and also wave equations with power nonlinearities).
  • Prove global existence and local well-posedness results for biharmonic wave maps (with rough initial data).

Results

During our study of wave equations with focusing power nonlinearity (which arise naturally as toy models for the wave maps equation) \[(\partial_t^2-\Delta_x)u(t,x)=u(t,x)^p\quad (t,x)\in \mathbb{R}\times\mathbb{R}^d\]we were able to show the existence of explicitly given self-similar blowup solutions for the quadratic and cubic wave equation (that is for $p=2$ and $p=3$) in all supercritical dimensions $d\geq 7$ and $d\geq 5$ respectively.
We furthermore analyzed their stability in the dimension $d=9$ and $d=7$ respectively:
Here, by exploiting the conformal invariance of the linearized equation, we found that a genuine unstable eigenvalue is given by $\lambda = 3$ (in both cases) and the corresponding eigenfunction can be computed explicitly from the time-translation symmetry mode. This allowed us to rigorously solve the spectral problem and to prove that the new solutions are co-dimension one stable (in a suitable sense) locally in backward light cones (see [GS21] and [CGS21]).

For biharmonic wave maps we showed that when the domain manifold $M$ is equal to $\mathbb{R}^m$ and $N = \mathbb{S}^{l-1}$ is a round sphere with the induced metric tensor that 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 (see [HLS19]).

Furthermore, the following results are from the PhD thesis of Tobias Schmid.

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).

Publications

  1. . Energy bounds for a fourth-order equation in low dimensions related to wave maps. Proc. Amer. Math. Soc., 151(1):225–237, January . URL https://doi.org/10.1090/proc/16100. [bibtex]

  2. and . Co-dimension one stable blowup for the supercritical cubic wave equation. Adv. Math., 390:107930, October . URL https://doi.org/10.1016/j.aim.2021.107930. [preprint] [bibtex]

  3. , , , and . Biharmonic wave maps: local wellposedness in high regularity. Nonlinearity, 33(5):2270–2305, March . URL https://doi.org/10.1088/1361-6544/ab73ce. [preprint] [bibtex]

  4. , , and . Threshold for blowup for the supercritical cubic wave equation. Nonlinearity, 33(5):2143–2158, March . URL https://doi.org/10.1088/1361-6544/ab6f4d. [preprint] [bibtex]

  5. and . Nonlinear asymptotic stability of homothetically shrinking Yang–Mills solitons in the equivariant case. Comm. Partial Differential Equations, 45(8):887–912, March . URL https://doi.org/10.1080/03605302.2020.1743308. [preprint] [bibtex]

  6. , , and . Biharmonic wave maps into spheres. Proc. Amer. Math. Soc., 148(2):787–796, August . URL https://doi.org/10.1090/proc/14744. [preprint] [bibtex]

Preprints

  1. , , and . On blowup for the supercritical quadratic wave equation. CRC 1173 Preprint 2021/40, Karlsruhe Institute of Technology, September . [bibtex]

  2. . Global results for a Cauchy problem related to biharmonic wave maps. CRC 1173 Preprint 2021/8, Karlsruhe Institute of Technology, February . [bibtex]

  3. . Energy bounds for biharmonic wave maps in low dimensions. CRC 1173 Preprint 2018/51, Karlsruhe Institute of Technology, December . [bibtex]

Theses

  1. . Local wellposedness and global regularity results for biharmonic wave maps. PhD thesis, Karlsruhe Institute of Technology (KIT), December . [bibtex]

Former staff
Name Title Function
Dr. Postdoctoral researcher
Prof. Dr. Member
M.Sc. Doctoral Researcher
Dr. Postdoctoral and doctoral researcher
Prof. Dr. Speaker of the IRTG
Prof. Dr. Member & scientific researcher
M.Sc. Doctoral researcher