The Zakharov–Kuznetsov equation in high dimensions: small initial data of critical regularity

Abstract

Abstract The Zakharov–Kuznetsov equation in spatial dimension $$d\ge 5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces, and it is proved that solutions scatter to free solutions as $$t \rightarrow \pm \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>→</mml:mo> <mml:mo>±</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . The proof is based on i) novel endpoint non-isotropic Strichartz estimates which are derived from the $$(d-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional Schrödinger equation, ii) transversal bilinear restriction estimates, and iii) an interpolation argument in critical function spaces. Under an additional radiality assumption, a similar result is obtained in dimension $$d=4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> .

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• Journal of Evolution Equations - View - PDF