$${\mathbb {L}}^p$$-solutions of deterministic and stochastic convective Brinkman–Forchheimer equations

Mohan, Manil T.

doi:10.1007/s13324-021-00595-0

Anal.Math.Phys.

2021

DOI: 10.1007/s13324-021-00595-0

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$${\mathbb {L}}^p$$-solutions of deterministic and stochastic convective Brinkman–Forchheimer equations

Manil T. Mohan

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Cited by 5 publications

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“…For deterministic CBF equations, the result on the existence of unique weak as well as strong solutions is established in [1,18,24,37,38], etc. For stochastic CBF equations, the existence of strong solutions (in the probabilistic sense) is obtained in [39,42] (Gaussian and pure jump noise), martingale solutions is established in [36,44] (Gaussian and Lévy noise), and mild solution is proved in [43] (Lévy noise). The existence of a unique pathwise strong solution for 3D stochastic NSE is a wellknown open problem, similarly it is open for 3D CBF equations for 1 ≤ r < 3.…”

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Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

Kinra

Mohan

2022

EECT

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This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in an <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional torus (<inline-formula><tex-math id="M2">\begin{document}$ n = 2, 3 $\end{document}</tex-math></inline-formula>):<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p = \boldsymbol{f}, \ \nabla\cdot\boldsymbol{u} = 0, $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M3">\begin{document}$ r\geq1 $\end{document}</tex-math></inline-formula>. We prove that the global attractor of the above system is singleton under small forcing intensity (<inline-formula><tex-math id="M4">\begin{document}$ r\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ r\geq 3 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ n = 3 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M8">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M9">\begin{document}$ r = n = 3 $\end{document}</tex-math></inline-formula>). But if one perturbs the above system with an additive or multiplicative white noise, there is no sufficient evidence that the random attractor keeps the singleton structure. We obtain that the random attractor for 2D stochastic CBF equations forced by additive and multiplicative white noise converges towards the deterministic singleton attractor for all <inline-formula><tex-math id="M10">\begin{document}$ 1\leq r<\infty $\end{document}</tex-math></inline-formula>, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). For the case of 3D stochastic CBF equations perturbed by additive and multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for <inline-formula><tex-math id="M11">\begin{document}$ 3\leq r<\infty $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M12">\begin{document}$ 2\beta\mu\geq 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M13">\begin{document}$ r = 3 $\end{document}</tex-math></inline-formula>), when the coefficient of random perturbation converges to zero.

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confidence: 99%

Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

Kinra

Mohan

2022

EECT

Self Cite

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Approximations of 2D and 3D Stochastic Convective Brinkman-Forchheimer Extended Darcy Equations

Mohan

2024

Acta Appl Math

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Existence and Uniqueness of Solutions to Backward 2D and 3D Stochastic Convective Brinkman–Forchheimer Equations Forced by Lévy Noise

Mohan

2023

Math Phys Anal Geom

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$${\mathbb {L}}^p$$-solutions of deterministic and stochastic convective Brinkman–Forchheimer equations

Cited by 5 publications

References 35 publications

Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

Approximations of 2D and 3D Stochastic Convective Brinkman-Forchheimer Extended Darcy Equations

Existence and Uniqueness of Solutions to Backward 2D and 3D Stochastic Convective Brinkman–Forchheimer Equations Forced by Lévy Noise

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