The Cauchy problem for wave maps

By Jalal Shatah, Michael Struwe. 2002

118 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/aff67dd307933b498ef2edae3498b350fae10ab4

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Global Solvability of the Cauchy Problem for the Landau-Lifshitz-Gilbert Equation in Higher Dimensions (2011, background): “Thus, in contrast to the present result for LLG, results for Schrödinger maps in higher dimensions take into account higher order (fractional) derivatives, similar to the case of wave maps, see [29, 32].”
Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions (2009, background): “There is also an extensive literature devoted to the more tractable higher dimensional case; we refer the reader to [31], [28], [8], [14], [17], and [20] for more information.”
Rough solutions for the wave maps equation (2005, methodology): “This includes work of Klainerman-Rodnianskii [10] (n ≥ 5), Nahmod-Stefanov-Uhlenbeck [14] and Shatah-Struwe [17] (n ≥ 4) and Krieger [12] (hyperbolic space, n = 3; also the case n = 2 is considered in a very recent manuscript).”
On the well-posedness of the wave map problem in high dimensions (2001, methodology): “e results on the modiﬁed wave map equation to show existence of wave maps is similar to the idea we used for non-linear Schrodinger [10] and not very diﬀerent from the technique used by Shatah-Struwe [11]. The translation depends on the compactness of M(or certain bounds on the isometric Nash embedd”
Global Regularity of Wave Maps from R2+1 to H2. Small Energy (2004, background): “(5) Massive simplification and extension of the previous case to n ≥ 4 by Shatah-Struwe [24].”
The wave maps equation (2004, background): “Klainerman and Rodnianski [13] did this for n ≥ 5, Shatah-Struwe [22] and Nahmod-Stefanov-Uhlenbeck [18] for n ≥ 4 and Krieger [15] for the hyperbolic space and n = 3.”
Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces (2011, background): “1 is closer to that of [38], [35], [28], and [29], in the sense that we prove a priori bounds on the derivatives of the Schrödinger map φ, in a suitable gauge, rather than the Schrödinger map itself.”
Global existence and uniqueness of Schrödinger maps in dimensions d... 4 (2007, methodology): “3 In the context of wave maps, orthonormal frames have been used in [4,14,18,21] etc.”
Low-regularity Schrödinger maps, II: global well-posedness in dimensions d ≥ 3 (2007, background): “Its direct analogue in the setting of wave maps is the work of Tataru [23] (see also [10], [12], [24], [21], [22], [11], [18], and [25] for other local and global well-posedness theorems for wave maps).”
Regularity of polyharmonic maps in the critical dimension (2009, methodology): “For Hölder continuity of u, our proof follows Wang’s proof [23] for the biharmonic case which is partially based on ideas by Shatah and Struwe [18].”
Uniqueness (2021, unknown): “…a simple yet general condition guaranteeing the uniqueness of critical points, which relies on an elementary observation: If I is a smooth functional and γ is a smooth curve connecting two critical points of I , then t (cid:2)→ I ( γ ( t )) =: F ( t ) ∈ R cannot be a strictly convex function.”