Remark on the optimal regularity for equations of wave maps type

Geometric renormalization of large energy wave maps

“…From Theorems 2.7 and 2.9 we see that the system (26) can be solved, and placed in the canonical heat-temporal gauge (29), given any Schwartz map φ : I × R 2 → N (as well as an orthonormal frame e(∞) in T φ(∞) N. In particular we can solve this equation for any Schwartz solution to the wave map equation (15). This gives rise to the fields φ, e, A α , ψ α , A s , ψ s on R + × I × R 2 solving the system of equations and boundary conditions…”

“…For higher dimensions n > 2 it is known that singularities can form 3 in finite time when the manifold N is positively curved, [46], [53], [5], [50] (and there is strong numerical evidence to suggest that there is also blowup when n = 2 [18], [2] in this case); when n ≥ 7 one can also obtain singularities for negatively curved manifolds [5], despite such manifolds being well-behaved for other equations (such as the harmonic map 2 The smoothness condition has been relaxed substantially, see for instance [26], [29], [21], [66], [67], [68], [69] and the discussion below; however we shall restrict our attention here to the Schwartz category. The rapid decay assumptions can also be removed by finite speed of propagation, though possibly at the cost of creating a domain of existence which is not a spacetime slab I × R n .…”

“…Geometrically, it is constructed by dragging the constant frame (φ(∞), e(∞)) back via parallel transport from s = +∞ to s = 0 by reversing the heat flow; it can also be viewed as the unique solution to (26) with boundary conditions (φ, e) = (φ(∞), e(∞)) when s = +∞ (ψ α , A α ) = 0 when s = +∞. (29) Note that we are specifying data at both s = 0 and s = +∞. As we shall explain shortly, this gauge is the analogue of the microlocal gauge in [64], [65], [27], [68] in the differentiated setting, and will be used here in place of the Coulomb gauge that was used in [38], [51], [31], [32], for reasons which will also be discussed shortly.…”

The Cauchy problem for wave maps on a curved background

“…In the non-equivariant case, Klainerman and Machedon in [11], [12], [13], [14], and Klainerman and Selberg in [16], [17], established strong well-posedness in the subcritical norm H s × H s−1 (R d ) with s > d 2 by exploiting the null-form structure present in (1.7).…”

On the continuity of the solution operator to the wave map system

“…This is classical if s is large enough, but for s close to the critical value s = n/2 it is a much more difficult result, due to Klainerman, Machedon, Selberg and obtained through careful bilinear estimates (see in particular [15]). See also Tataru [29] for the case of Besov spaces.…”