Bernis estimates for higher-dimensional doubly-degenerate non-Newtonian thin-film equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915073554907136 |
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| author | Lienstromberg, Christina Nik, Katerina |
| author_facet | Lienstromberg, Christina Nik, Katerina |
| contents | For the doubly-degenerate parabolic non-Newtonian thin-film equation $$ u_t + \text{div}\bigl(u^n |\nabla Δu|^{p-2} \nabla Δu\bigr) = 0, $$ we derive (local versions) of Bernis estimates of the form $$
\int_Ω u^{n-2p} |\nabla u|^{3p}\, dx
+
\int_Ω u^{n-\frac{p}{2}} |Δu|^{\frac{3p}{2}}\, dx
\leq c(n,p,d) \int_Ω u^n|\nabla Δu|^p\, dx, $$ for functions $u \in W^2_p(Ω)$ with Neumann boundary condition, where $2 \leq p < \frac{19}{3}$ and $n$ lies in a certain range. Here, $Ω\subset \mathbb{R}^d$ is a smooth convex domain with $d < 3p$. A particularly important consequence is the estimate $$
\int_Ω
|\nabla Δ(u^{\frac{n+p}{p}})|^p\, dx \leq c(n,p,d) \int_Ω u^n|\nabla Δu|^p\, dx. $$ The methods used in this article follow the approach of [Grü01] for the Newtonian case, while addressing the specific challenges posed by the nonlinear higher-order term $|\nabla Δu|^{p-2} \nabla Δu$ and the additional degeneracy. The derived estimates are key to establishing further qualitative results, such as the existence of weak solutions, finite propagation of support, and the appearance of a waiting-time phenomenon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15883 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bernis estimates for higher-dimensional doubly-degenerate non-Newtonian thin-film equations Lienstromberg, Christina Nik, Katerina Analysis of PDEs 76A05, 76A20, 35A23, 46B70, 35Q35, 35K35, 35K65 For the doubly-degenerate parabolic non-Newtonian thin-film equation $$ u_t + \text{div}\bigl(u^n |\nabla Δu|^{p-2} \nabla Δu\bigr) = 0, $$ we derive (local versions) of Bernis estimates of the form $$ \int_Ω u^{n-2p} |\nabla u|^{3p}\, dx + \int_Ω u^{n-\frac{p}{2}} |Δu|^{\frac{3p}{2}}\, dx \leq c(n,p,d) \int_Ω u^n|\nabla Δu|^p\, dx, $$ for functions $u \in W^2_p(Ω)$ with Neumann boundary condition, where $2 \leq p < \frac{19}{3}$ and $n$ lies in a certain range. Here, $Ω\subset \mathbb{R}^d$ is a smooth convex domain with $d < 3p$. A particularly important consequence is the estimate $$ \int_Ω |\nabla Δ(u^{\frac{n+p}{p}})|^p\, dx \leq c(n,p,d) \int_Ω u^n|\nabla Δu|^p\, dx. $$ The methods used in this article follow the approach of [Grü01] for the Newtonian case, while addressing the specific challenges posed by the nonlinear higher-order term $|\nabla Δu|^{p-2} \nabla Δu$ and the additional degeneracy. The derived estimates are key to establishing further qualitative results, such as the existence of weak solutions, finite propagation of support, and the appearance of a waiting-time phenomenon. |
| title | Bernis estimates for higher-dimensional doubly-degenerate non-Newtonian thin-film equations |
| topic | Analysis of PDEs 76A05, 76A20, 35A23, 46B70, 35Q35, 35K35, 35K65 |
| url | https://arxiv.org/abs/2412.15883 |