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<p>We consider the Cauchy problem for the incompressible Navier–Stokes equations in dimension three and construct initial data in the critical space <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>B</mi> <mi>M</mi> <msup> <mi>O</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$BMO^{-1}$</EquationSource> </InlineEquation> from which there exist two distinct global solutions, both smooth for all <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>t</mi> <mo>&gt;</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$t&gt;0$</EquationSource> </InlineEquation>. One consequence of this construction is the sharpness of the celebrated small data global well-posedness result of Koch and Tataru. This appears to be the first example of non-uniqueness for the Navier–Stokes equations with data at the critical regularity. The proof is based on a non-uniqueness mechanism proposed by the second author in the context of the dyadic Navier–Stokes equations.</p>

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Non-uniqueness of smooth solutions of the Navier–Stokes equations from critical data

  • 期刊论文

  • 发表日期：

    2025年12月12日

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  • Matei P. Coiculescu,
  • Stan Palasek

摘要

We consider the Cauchy problem for the incompressible Navier–Stokes equations in dimension three and construct initial data in the critical space B M O − 1 $BMO^{-1}$ from which there exist two distinct global solutions, both smooth for all t > 0 $t>0$ . One consequence of this construction is the sharpness of the celebrated small data global well-posedness result of Koch and Tataru. This appears to be the first example of non-uniqueness for the Navier–Stokes equations with data at the critical regularity. The proof is based on a non-uniqueness mechanism proposed by the second author in the context of the dyadic Navier–Stokes equations.