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The Cauchy problem for wave maps. (English) Zbl 1024.58014

From the text: Let \((N,h)\) be a complete Riemannian manifold of dimension \(k\) with \(\partial N=\varnothing\). Denoting space-time coordinates on \(\mathbb{R}^{m+1}\) as \((t,x)= (x^\alpha)\), \(0\leq \alpha\leq m\), a wave map \(u: \mathbb{R}^{m+1}\to N\) is a solution to the equation \[ D^\alpha \partial_\alpha u= 0,\tag{1} \] where \(\partial_\alpha= \partial/\partial x^\alpha\) and where we raise and lower indices with the Minkowski metric \((n_{\alpha\beta})= \text{diag}(- 1,1,\dots, 1)\). We tacitly sum over repeated indices. Moreover, \(D\) is the covariant pullback derivative in the bundle \(u^*TN\).
Next to the intrinsic form (1) of the wave map equation, the following extrinsic representation will be useful. Recall that the Nash embedding theorem permits to regard \(N\) as a submanifold of some Euclidean space \(\mathbb{R}^n\). Letting \(u= (u^1,\dots, u^n): \mathbb{R}^{m+1}\to N\hookrightarrow \mathbb{R}^n\) be the corresponding extrinsic representation of our wave map \(u\), (1) then takes the form \[ u^i_{tt}-\Delta u^i= B^i_{jk}(u) (\partial_\alpha u^j, \partial^\alpha u^k),\qquad 1\leq i\leq n, \] where \(B(p): T_pN\times T_pN\to (T_pN)^\perp\) is the second fundamental of \(N\subset \mathbb{R}^n\) at any \(p\in N\).
In this paper, we study the Cauchy problem for wave maps with initial data \[ (u, u_t)|_{t=0}= (u_0, u_1)\in H^{m/2}\times H^{m/2- 1}(\mathbb{R}^m; TN).\tag{2} \] Our main result may be stated as follows.
Theorem 1.1. Suppose \(N\) is complete, without boundary and has bounded curvature in the sense that the curvature operator \(R\) and the second fundamental form \(B\) and all their derivatives are bounded, and let \(m\geq 4\). Then there is a constant \(\varepsilon_0> 0\) such that for any \((u_0, u_1)\in H^{m/2}\times H^{m/2-1}(\mathbb{R}^m; TN)\) satisfying \[ \|u_0\|_{\dot H^{m/2}}+\|u_1\|_{\dot H^{m/2-1}}< \varepsilon_0, \] there exists a unique global solution \(u\in C^0(\mathbb{R}; H^{m/2})\cap C^1(\mathbb{R}; H^{m/2-1})\) of (1) and (2) satisfying \[ \sup_t\|du(t)\|_{\dot H^{m/2-1}}+ \int_{\mathbb{R}}\|du(t)\|^2_{L^{2m}(\mathbb{R}^m)} dt\leq C\varepsilon_0 \] and preserving any higher regularity of the data. Here \(\dot H^s\) denotes the homogeneous Sobolev space.

MSC:

58J47 Propagation of singularities; initial value problems on manifolds
35L70 Second-order nonlinear hyperbolic equations
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