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Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability

UDC 519.21 We initiate the investigation of stochastic Navier–Stokes variational inequalities involving unilateral boundary conditions and nonlinear forcings driven by Wiener processes for which we establish the existence of a probabilistic weak (or martingale) soluti...

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Date:2023
Main Author: Sango, M.
Format: Article
Language:English
Published: Institute of Mathematics, NAS of Ukraine 2023
Online Access:https://umj.imath.kiev.ua/index.php/umj/article/view/6665
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Journal Title:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
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author Sango, M.
Sango, M.
author_facet Sango, M.
Sango, M.
author_sort Sango, M.
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
collection OJS
datestamp_date 2023-05-14T15:30:06Z
description UDC 519.21 We initiate the investigation of stochastic Navier–Stokes variational inequalities involving unilateral boundary conditions and nonlinear forcings driven by Wiener processes for which we establish the existence of a probabilistic weak (or martingale) solution.  Our approach involves an intermediate penalized problem whose weak solution is obtained by means of Galerkin's method in combination with some analytic and probabilistic compactness results.  The required probabilistic weak solution of the stochastic Navier–Stokes variational inequality is consecutively obtained through the limit transition in the penalized problem. The main result is new for stochastic Navier–Stokes variational inequalities. It is a stochastic counterpart of the work of Brezis on deterministic Navier–Stokes variational inequalities and generalizes several previous results on stochastic Navier-Stokes equations to stochastic Navier–Stokes variational inequalities with unilateral boundary conditions.
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spelling umjimathkievua-article-66652023-05-14T15:30:06Z Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability Sango, M. Sango, M. NAVIER – STOKES VARIATIONAL INEQUALITIES UDC 519.21 We initiate the investigation of stochastic Navier–Stokes variational inequalities involving unilateral boundary conditions and nonlinear forcings driven by Wiener processes for which we establish the existence of a probabilistic weak (or martingale) solution.  Our approach involves an intermediate penalized problem whose weak solution is obtained by means of Galerkin's method in combination with some analytic and probabilistic compactness results.  The required probabilistic weak solution of the stochastic Navier–Stokes variational inequality is consecutively obtained through the limit transition in the penalized problem. The main result is new for stochastic Navier–Stokes variational inequalities. It is a stochastic counterpart of the work of Brezis on deterministic Navier–Stokes variational inequalities and generalizes several previous results on stochastic Navier-Stokes equations to stochastic Navier–Stokes variational inequalities with unilateral boundary conditions. УДК 519.21 Стохастичні варіаційні нерівності Нав'є–Стокса з односторонніми граничними умовами: ймовірнісна слабка розв’язність У цій статті розпочато вивчення стохастичних варіаційних нерівностей  Нав’є–Стокса з односторонніми граничними умовами та нелінійними впливами, що викликані вінеровськими процесами, для яких встановлюється існування ймовірнісного слабкого (або мартингального) розв'язку.  Наш підхід включає проміжну штрафну задачу, слабкий розв'язок якої отримано за допомогою методу Гальоркіна в поєднанні з деякими аналітичними та ймовірнісними результатами щодо компактності. Шуканий імовірнісний слабкий розв’язок стохастичної варіаційної нерівності Нав’є–Стокса одержано в результаті граничного переходу в штрафній задачі. Отриманий основний результат є новим для стохастичних варіаційних нерівностей Нав’є–Стокса. Він є стохастичним аналогом роботи Брезіса щодо детермінованих варіаційних нерівностей Нав’є–Стокса та узагальнює кілька попередніх результатів, отриманих для стохастичних рівнянь Нав’є–Стокса, на випадок стохастичних варіаційних нерівностей Нав’є–Стокса з односторонніми граничними умовами. Institute of Mathematics, NAS of Ukraine 2023-05-10 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/6665 10.37863/umzh.v75i4.6665 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 4 (2023); 523 - 541 Український математичний журнал; Том 75 № 4 (2023); 523 - 541 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6665/9783 Copyright (c) 2023 M SANGO
spellingShingle Sango, M.
Sango, M.
Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability
title Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability
title_alt Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability
title_full Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability
title_fullStr Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability
title_full_unstemmed Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability
title_short Stochastic Navier–Stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability
title_sort stochastic navier–stokes variational inequalities with unilateral boundary conditions: probabilistic weak solvability
topic_facet NAVIER – STOKES VARIATIONAL INEQUALITIES
url https://umj.imath.kiev.ua/index.php/umj/article/view/6665
work_keys_str_mv AT sangom stochasticnavierstokesvariationalinequalitieswithunilateralboundaryconditionsprobabilisticweaksolvability
AT sangom stochasticnavierstokesvariationalinequalitieswithunilateralboundaryconditionsprobabilisticweaksolvability

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