Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions
By Aynur Bulut, M. Czubak, Dong Li, Natavsa Pavlovi'c, Xiaoyi Zhang. 2009
55 citations recorded by the source corpus.
In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ℝ × ℝ d with d ≥ 6. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the analysis of Kenig and Merle [17] of the dynamics of the focusing (NLW) below the energy threshold. By employing refined paraproduct estimates we also prove unconditional uniqueness of solutions for d ≥ 6 in the natural energy class. This extends an earlier result by Planchon [26].
Subfields: Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e6df468a0e707e7ce63cd91a0152da89c974c4bd
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In Mathematical Physics
Citations in the corpus
- Asymptotic decomposition for semilinear Wave and equivariant wave map equations (2015, background): “The extension to higher equivariance classes is straightforward, with the help of the recent works [3, 31] where the more sophisticated Cauchy theory for higher dimensional energy critical wave equation (involving fractional power nonlinearities) has been developed.”
- Global Well-Posedness and Scattering for the Defocusing Energy-Supercritical Cubic Nonlinear Wave Equation (2010, background): “l-posedness theorem for (NLW) with our cubic nonlinearity F(u) = |u|2u. The version that we present here is in the spirit of the related results in the works of [19, 25]. For similar results see also [3, 7, 18, 32, 35, 41]. Theorem 4.4. Let d≥ 6 and sc = d−2 2 . Then for all A>0, there exists δ 0”