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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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