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Sharp estimates for linear and nonlinear wave equations via the Penrose transform

Abstract : We apply the Penrose transform, which is a basic tool of relativistic physics, to the study of sharp estimates for linear and nonlinear wave equations. We disprove a conjecture of Foschi, regarding extremizers for the Strichartz inequality with data in the Sobolev space Ḣ½ x Ḣ⁻½ (Rᵈ), for even d ⩾2. On the other hand, we provide evidence to support the conjecture in odd dimensions and refine his sharp inequality in R¹⁺³, adding a term proportional to the distance of the initial data from the set of extremizers. Using this, we provide an asymptotic formula for the Strichartz norm of small solutions to the cubic wave equation in Minkowski space. The leading coefficient is given by Foschi’s sharp constant. We calculate the constant in the second term, whose absolute value and sign changes depending on whether the equation is focusing or defocusing.
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Giuseppe Negro. Sharp estimates for linear and nonlinear wave equations via the Penrose transform. Analysis of PDEs [math.AP]. Université Sorbonne Paris Cité; Universidad autonóma de Madrid, 2018. English. ⟨NNT : 2018USPCD071⟩. ⟨tel-02619753⟩

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