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On the regularity properties of a model problem related to wave maps. (English) Zbl 0878.35075
We consider the nonlinear wave equation (1) $$\square \varphi =\partial_i [\varphi,R_0R_i \varphi]$$ subject to the initial value problem $\varphi(0,x) =f_0(x)\in H^s(\mathbb{R}^n) \quad\partial_t \varphi(0,x) =g_0(x) \in H^{s-1} (\mathbb{R}^n). \tag{2}$ Here $$\varphi$$ is a scalar function defined on Minkowski space-time $$\mathbb{R}^{n+1}$$ with values in a fixed Lie algebra of matrices endowed with the standard Lie bracket $$[\varphi,\psi] =\varphi \cdot \psi-\psi \cdot\varphi$$. The operators $$R_0$$, $$R_i$$ are the nonlocal operators $$R_0= (-\Delta)^{-1/2} \partial_t$$, $$R_i= (-\Delta)^{-1/2} \partial_i$$.
The classical local existence theorem applied to (1) requires $$s>n/2$$. The optimal result, however, should only require $$s> (n-2)/2$$. One also expects that if the data are in $$H^s$$ for $$s>(n-2)/2$$ and sufficiently small in the homogeneous Sobolev norm $$\dot H^{(n-2)/2}$$, then the corresponding solutions can be extended for all time. Our long-term goal is to prove the later statement by Fourier analysis techniques and then try to remove the smallness assumption in the critical case $$n=2$$ by a different type of argument. In this paper, we can only prove the first statement mentioned above, namely, the following: The initial value problem (1), (2) is well posed for $$s>(n-2)/2$$.

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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##### References:
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