Energy dispersed large data wave maps in \(2+1\) dimensions. (English) Zbl 1218.35129

The authors consider finite energy large data wave maps from the Minkowski space \(\mathbb R^{2+1}\) into a compact Riemannian manifold \((M,g)\).
The Cauchy problem for the wave map equation has the form
\[ (\partial_t^2-\Delta)\varphi^a=-S^a_{bc}(\varphi)\partial_\alpha \varphi^b \partial^\alpha \varphi^c, \quad\varphi\in\mathbb R^N, \]
with initial data \(\varphi(0,x)=\varphi_0(x)\), \(\partial_t \varphi(0,x)=\varphi_1(x)\) obeying the constraints \(\varphi_0(x)\in M\), \(\varphi_1(x)\in T_{\varphi_0(x)}M\), \(x\in\mathbb R^2\). The system admits a conserved energy,
\[ E[\varphi](t):=\int_{\mathbb R^2}(|\partial_t \varphi|^2+|\nabla_x \varphi|^2)\,dx=E. \]
The main result of the paper is a conditional regularity theorem, which states that there exist two functions \(0<\varepsilon(E)\ll 1\ll F(E)\) of the energy such that the following statement is true. If \(\varphi\) is a finite energy solution on the open interval \((t_1,t_2)\) with energy \(E\) and \(\sup_k\|P_k \varphi\|_{L^\infty_{t,x}[(t_1,t_2)\times\mathbb R^2]}\leq \varepsilon(E)\), then one also has \(\|\varphi\|_{S(t_1,t_2)}\leq F(E)\), and such a solution \(\varphi(t)\) extends in a regular way to a neighborhood of the interval \([t_1,t_2]\), where \(S\) is a variant of the standard dispersive norm from D. Tataru [Am. J. Math. 127, No. 2, 293–377 (2005; Zbl 1330.58021)]. The authors also prove a weaker non-conditional version of this result as a consequence of its frequency envelope version.


35L15 Initial value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
58J45 Hyperbolic equations on manifolds


Zbl 1330.58021
Full Text: DOI arXiv


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