<p>In this article, we show the necessary and sufficient conditions for the inequality <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Vert u \Vert_{{{L}_{t}^{q}}{{L}_{x}^{r}}} \underset{\sim}{<} \Vert u \Vert_{{{X}^{s,b}}}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo fence="false" stretchy="false">∥</mo> <mi>u</mi> <msub> <mo>∥</mo> <mrow> <mrow> <msubsup> <mrow> <mi>L</mi> </mrow> <mrow> <mi>t</mi> </mrow> <mrow> <mi>q</mi> </mrow> </msubsup> </mrow> <mrow> <msubsup> <mrow> <mi>L</mi> </mrow> <mrow> <mi>x</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msubsup> </mrow> </mrow> </msub> <munder> <mo><</mo> <mo>∼</mo> </munder> <mo>∥</mo> <mi>u</mi> <msub> <mo fence="false" stretchy="false">∥</mo> <mrow> <mrow> <msup> <mrow> <mi>X</mi> </mrow> <mrow> <mi>s</mi> <mo>,</mo> <mi>b</mi> </mrow> </msup> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation>, where <Equation ID="Equa"> <EquationSource Format="TEX">$\Vert u \Vert_{{{X}^{s,b}}} := {\Vert{\hat u}(\tau, \; \xi) \langle \xi \rangle^{s} \langle \tau - \xi^{3}\rangle^{b}\Vert}_{{L}_{\tau, \xi}^{2}}.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo fence="false" stretchy="false">∥</mo> <mi>u</mi> <msub> <mo>∥</mo> <mrow> <mrow> <msup> <mrow> <mi>X</mi> </mrow> <mrow> <mi>s</mi> <mo>,</mo> <mi>b</mi> </mrow> </msup> </mrow> </mrow> </msub> <mo>:=</mo> <msub> <mrow> <mo fence="false" stretchy="false">∥</mo> <mrow> <mrow> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>ξ</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow> <mi>s</mi> </mrow> </msup> <mo fence="false" stretchy="false">⟨</mo> <mi>τ</mi> <mo>−</mo> <msup> <mi>ξ</mi> <mrow> <mn>3</mn> </mrow> </msup> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow> <mi>b</mi> </mrow> </msup> <mo fence="false" stretchy="false">∥</mo> </mrow> <mrow> <msubsup> <mrow> <mi>L</mi> </mrow> <mrow> <mi>τ</mi> <mo>,</mo> <mi>ξ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </msub> <mo>.</mo> </math></EquationSource> </Equation></p><p>Here, we provide a complete classification of the indices relationships for which this inequality holds true. Such estimates will be very useful in solving the well-posedness for low regularity well-posedness of the Korteweg-de Vries equations and stochastic Korteweg-de Vries equations.</p>

Strichartz Type Estimates of the Airy Equation

期刊论文
发表日期：

2026年7月7日

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Jie Chen,
Fan Gu,
Bo-ling Guo

摘要

In this article, we show the necessary and sufficient conditions for the inequality $\Vert u \Vert_{{{L}_{t}^{q}}{{L}_{x}^{r}}} \underset{\sim}{<} \Vert u \Vert_{{{X}^{s,b}}}$ $∥ u ∥_{L_{t}^{q} L_{x}^{r}} \underset{\sim}{<} ∥ u ∥_{X^{s, b}}$ , where $\Vert u \Vert_{{{X}^{s,b}}} := {\Vert{\hat u}(\tau, \; \xi) \langle \xi \rangle^{s} \langle \tau - \xi^{3}\rangle^{b}\Vert}_{{L}_{\tau, \xi}^{2}}.$ $∥ u ∥_{X^{s, b}} := {∥ \hat{u} (τ, ξ) ⟨ ξ ⟩^{s} ⟨ τ - ξ^{3} ⟩^{b} ∥}_{L_{τ, ξ}^{2}} .$

Here, we provide a complete classification of the indices relationships for which this inequality holds true. Such estimates will be very useful in solving the well-posedness for low regularity well-posedness of the Korteweg-de Vries equations and stochastic Korteweg-de Vries equations.