## 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.

### MSC:

 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:

### References:

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