On the convective Brinkman-Forchheimer equations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915085624016896 |
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| author | Gautam, Sagar Mohan, Manil T. |
| author_facet | Gautam, Sagar Mohan, Manil T. |
| contents | The convective Brinkman--Forchheimer equations or the Navier--Stokes equations with damping in bounded or periodic domains $\subset\mathbb{R}^d$, $2\leq d\leq 4$ are considered in this work. The existence and uniqueness of a global weak solution in the Leray-Hopf sense satisfying the energy equality to the system: $$\partial_t\boldsymbol{u}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\ \nabla\cdot\boldsymbol{u}=0,$$ (for all values of $β>0$ and $μ>0$, whenever the absorption exponent $r>3$ and $2βμ\geq 1$, for the critical case $r=3$) is proved. We exploit the monotonicity as well as the demicontinuity properties of the linear and nonlinear operators and the Minty-Browder technique in the proofs. Finally, we discuss the existence of global-in-time strong solutions to such systems in periodic domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20940 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the convective Brinkman-Forchheimer equations Gautam, Sagar Mohan, Manil T. Analysis of PDEs The convective Brinkman--Forchheimer equations or the Navier--Stokes equations with damping in bounded or periodic domains $\subset\mathbb{R}^d$, $2\leq d\leq 4$ are considered in this work. The existence and uniqueness of a global weak solution in the Leray-Hopf sense satisfying the energy equality to the system: $$\partial_t\boldsymbol{u}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\ \nabla\cdot\boldsymbol{u}=0,$$ (for all values of $β>0$ and $μ>0$, whenever the absorption exponent $r>3$ and $2βμ\geq 1$, for the critical case $r=3$) is proved. We exploit the monotonicity as well as the demicontinuity properties of the linear and nonlinear operators and the Minty-Browder technique in the proofs. Finally, we discuss the existence of global-in-time strong solutions to such systems in periodic domains. |
| title | On the convective Brinkman-Forchheimer equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.20940 |