Non‐uniqueness of Leray–Hopf solutions to the forced fractional Navier–Stokes equations in three dimensions, up to the J. L. Lions exponent

Khor, Calvin; Miao, Changxing; Su, Xiaoyan

doi:10.1112/blms.12889

Cited by 1 publication

(5 citation statements)

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“…Theorem 3.10 below provides the crucial instability of velocity operators. See also [53]. Below we give the more detailed proof.…”

Section: Instability Of Vorticity and Velocity Operatorsmentioning

confidence: 94%

“…Therefore, the (hyper-viscous) component can be treated perturbatively. See also the very recent work [53].…”

Section: Strategy Of Proofmentioning

confidence: 94%

“…The linear operator L ss is mainly studied in Subsections 3.1 and 3.2 by means of the theory of semigroups and operators, based on the Hille-Yosida theorem and stability theory of semigroups [25,44]. This approach to the problem is novel and partially deviates from the analysis in [53]. The second solution.…”

Section: Strategy Of Proofmentioning

confidence: 99%

“…By virtue of Propositions 3.3 and 3.4 and the compactness of operator K, one has the invertibility and convergence results below. See [1,53] for relevant arguments. Lemma 3.5 (Resolvents of T β ).…”

Section: Instability Of Vorticity and Velocity Operatorsmentioning

confidence: 99%

“…As a consequence, in view of the Cauchy Theorem and uniform convergence in Proposition 3.4, one has the key instability of the vorticity operator L (β) vor . See also [53].…”

Section: Instability Of Vorticity and Velocity Operatorsmentioning

confidence: 99%

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Non-uniqueness of Leray solutions of the forced Navier-Stokes equations

2022

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This paper concerns the forced stochastic Navier-Stokes equation driven by additive noise in the three dimensional Euclidean space. By constructing an appropriate forcing term, we prove that there exist distinct Leray solutions in the probabilistically weak sense. In particular, the joint uniqueness in law fails in the Leray class. The non-uniqueness also displays in the probabilistically strong sense in the local time regime, up to stopping times. Furthermore, we discuss the optimality from two different perspectives: sharpness of the hyper-viscous exponent and size of the external force. These results in particular yield that the Lions exponent is the sharp viscosity threshold for the uniqueness/non-uniqueness in law of Leray solutions. Our proof utilizes the self-similarity and instability programme developed by 43] and Albritton-Brué-Colombo [1], together with the theory of martingale solutions including stability for non-metric spaces and gluing procedure.

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“…Theorem 3.10 below provides the crucial instability of velocity operators. See also [53]. Below we give the more detailed proof.…”

Section: Instability Of Vorticity and Velocity Operatorsmentioning

confidence: 94%

“…Therefore, the (hyper-viscous) component can be treated perturbatively. See also the very recent work [53].…”

Section: Strategy Of Proofmentioning

confidence: 94%

Section: Strategy Of Proofmentioning

confidence: 99%

Section: Instability Of Vorticity and Velocity Operatorsmentioning

confidence: 99%

“…As a consequence, in view of the Cauchy Theorem and uniform convergence in Proposition 3.4, one has the key instability of the vorticity operator L (β) vor . See also [53].…”

Section: Instability Of Vorticity and Velocity Operatorsmentioning

confidence: 99%

See 3 more Smart Citations