Higher-order nonlinear dispersive equations

Abstract

We study nonlinear dispersive equations of the form <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript t Baseline u plus partial-differential Subscript x Superscript 2 j plus 1 Baseline u plus upper P left-parenthesis u comma partial-differential Subscript x Baseline u comma ellipsis comma partial-differential Subscript x Superscript 2 j Baseline u right-parenthesis equals 0 comma x comma t element-of double-struck upper R comma j element-of double-struck upper Z Superscript plus Baseline comma"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi mathvariant="normal">∂<!-- ∂ --></mml:mi> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>j</mml:mi> </mml:mrow> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="2em" /> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width="1em" /> <mml:mi>j</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\partial _t}u + \partial _x^{2j + 1}u + P(u,{\partial _x}u, \ldots ,\partial _x^{2j}u) = 0,\qquad x,t \in \mathbb {R},\quad j \in {\mathbb {Z}^ + },</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis dot right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">P( \cdot )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a polynomial having no constant or linear terms. It is shown that the associated initial value problem is locally well posed in weighted Sobolev spaces. The method of proof combines several sharp estimates for solutions of the associated linear problem and a change of dependent variable which allows us to consider data of arbitrary size.

Locations

• Proceedings of the American Mathematical Society - View - PDF