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Global martingale weak solutions for the three-dimensional stochastic chemotaxis-Navier-Stokes system with L\'{e}vy processes
Analysis of PDEs · Submitted 14 Sep 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove in present work that the three-dimensional SCNS system possesses at least one global martingale solution under proper assumptions, which is weak both in the analytical sense and in the stochastic sense.
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The abstract, sentence by sentence
This paper studies the three-dimensional stochastic chemotaxis-Navier-Stokes (SCNS) system subjected to a L\'{e}vy-type random external force in bounded domain. Up to now, the existing results concerning global solvability of SCNS system mainly concentrated on the case of two spacial dimensions, little is known for the SCNS system in dimension three. We prove in present work that the three-dimensional SCNS system possesses at least one global martingale solution under proper assumptions, which is weak both in the analytical sense and in the stochastic sense. A new stochastic analogue of entropy-energy inequality and an uniform boundedness estimate are derived, which enable us to construct global-in-time approximate solutions from a properly regularized SCNS system via the Contraction Mapping Principle. The proof of the existence of martingale solution is based on the stochastic compactness method and an elaborate identification of the limits procedure, where the Jakubowski-Skorokhod Theorem is applied to deal with the phase spaces equipped with weak topology.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper studies the three-dimensional stochastic chemotaxis-Navier-Stokes (SCNS) system subjected to a L\'{e}vy-type random external force in bounded domain. | background 0.55 | 0.48 | 0.03 | asserted 0.97 |
| s1 | Up to now, the existing results concerning global solvability of SCNS system mainly concentrated on the case of two spacial dimensions, little is known for the SCNS system in dimension three. | gap 0.97 | 0.03 | 0.02 | asserted 1.00 |
| s2 | We prove in present work that the three-dimensional SCNS system possesses at least one global martingale solution under proper assumptions, which is weak both in the analytical sense and in the stochastic sense. | result 0.85 | 0.97 | 0.96 | asserted 1.00 |
| s3 | A new stochastic analogue of entropy-energy inequality and an uniform boundedness estimate are derived, which enable us to construct global-in-time approximate solutions from a properly regularized SCNS system via the Contraction Mapping Principle. | method 0.86 | 0.97 | 0.05 | asserted 1.00 |
| s4 | The proof of the existence of martingale solution is based on the stochastic compactness method and an elaborate identification of the limits procedure, where the Jakubowski-Skorokhod Theorem is applied to deal with the phase spaces equipped with weak topology. | method 1.00 | 0.91 | 0.06 | asserted 1.00 |
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Original abstract
This paper studies the three-dimensional stochastic chemotaxis-Navier-Stokes (SCNS) system subjected to a L\'{e}vy-type random external force in bounded domain. Up to now, the existing results concerning global solvability of SCNS system mainly concentrated on the case of two spacial dimensions, little is known for the SCNS system in dimension three. We prove in present work that the three-dimensional SCNS system possesses at least one global martingale solution under proper assumptions, which is weak both in the analytical sense and in the stochastic sense. A new stochastic analogue of entropy-energy inequality and an uniform boundedness estimate are derived, which enable us to construct global-in-time approximate solutions from a properly regularized SCNS system via the Contraction Mapping Principle. The proof of the existence of martingale solution is based on the stochastic compactness method and an elaborate identification of the limits procedure, where the Jakubowski-Skorokhod Theorem is applied to deal with the phase spaces equipped with weak topology.
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2 arXiv papers cite this in 3 sentences
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One of the most important contribution of this work is to consider the stochastic Navier-Stokes equations [ref] with a fractional dissipation $(-Δ)^αu$, by extending the previous works [ref] .
Random perturbations for the chemotaxis-fluid model with fractional dissipation: Global pathwise weak solutions · Introduction · 1.00
What it cites
Cites as background 1 paper
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In this aspect, we would like to mention the recent breakthrough on the non-uniqueness of weak solutions to the deterministic Navier-Stokes equations [ref] and to the stochastic Navier-Stokes equations [ref] based on the convex integration method.
Sharp non-uniqueness of solutions to stochastic Navier-Stokes equations · Main result · 0.81
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