Nondispersive solutions to the L
2-critical Half-Wave Equation |
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Authors: | Joachim Krieger Enno Lenzmann Pierre Raphaël |
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Institution: | 1. Department of Mathematics, EPFL, Lausanne, 1051, Switzerland 2. Mathematisches Institut, Universit?t Basel, Rheinsprung 21, Basel, 4051, Switzerland 3. Laboratoire J.A. Dieudonné, UMR CNRS 7351, Université de Nice Sophia-Antipolis, Parc Valrose, 06108, NICE Cedex 02, France
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Abstract: | We consider the focusing L 2-critical half-wave equation in one space dimension, $$i \partial_t u = D u - |u|^2 u$$ , where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold ${M_{*} > 0}$ such that all H 1/2 solutions with ${\|u\|_{L^2} < M_*}$ extend globally in time, while solutions with ${\|u\|_{L^2} \geq M_*}$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass ${\|u_0\|_{L^2} = M_*}$ . More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E 0 > 0 and the linear momentum ${P_0 \in \mathbb{R}}$ . In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L 2-critical nonlinear PDEs with nonlocal dispersion. |
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