A fast regularisation of a Newtonian vortex equation
We consider equations of the form $ u_t = \nabla \cdot ( \gamma(u) \nabla \mathrm{N}(u))$, where $\mathrm{N}$ is the Newtonian potential (inverse of the Laplacian) posed in the whole space $\mathbb R^d$, and $\gamma(u)$ is the mobility. For linear mobility, $\gamma(u)=u$, the equation and some varia...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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EMS Press
2022
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| _version_ | 1826307607816568832 |
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| author | Carrillo, JA Gómez-Castro, D Vázquez, JL |
| author_facet | Carrillo, JA Gómez-Castro, D Vázquez, JL |
| author_sort | Carrillo, JA |
| collection | OXFORD |
| description | We consider equations of the form $ u_t = \nabla \cdot ( \gamma(u) \nabla
\mathrm{N}(u))$, where $\mathrm{N}$ is the Newtonian potential (inverse of the
Laplacian) posed in the whole space $\mathbb R^d$, and $\gamma(u)$ is the
mobility. For linear mobility, $\gamma(u)=u$, the equation and some variations
have been proposed as a model for superconductivity or superfluidity. In that
case the theory leads to uniqueness of bounded weak solutions having the
property of compact space support, and in particular there is a special
solution in the form of a disk vortex of constant intensity in space
$u=c_1t^{-1}$ supported in a ball that spreads in time like $c_2t^{1/d}$, thus
showing a discontinuous leading front.
In this paper we propose the model with sublinear mobility
$\gamma(u)=u^\alpha$, with $0<\alpha<1$, and prove that nonnegative solutions
recover positivity everywhere, and moreover display a fat tail at infinity. The
model acts in many ways as a regularization of the previous one. In particular,
we find that the equivalent of the previous vortex is an explicit self-similar
solution decaying in time like $u=O(t^{-1/\alpha})$ with a space tail with size
$u=O(|x|^{- d/(1-\alpha)})$. We restrict the analysis to radial solutions and
construct solutions by the method of characteristics. We introduce the mass
function, which solves an unusual variation of Burger's equation, and plays an
important role in the analysis. We show well-posedness in the sense of
viscosity solutions. We also construct numerical finite-difference convergent
schemes. |
| first_indexed | 2024-03-07T07:07:08Z |
| format | Journal article |
| id | oxford-uuid:03b5a5ab-e7bd-4568-9052-d1c84a06b78e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:07:08Z |
| publishDate | 2022 |
| publisher | EMS Press |
| record_format | dspace |
| spelling | oxford-uuid:03b5a5ab-e7bd-4568-9052-d1c84a06b78e2022-05-16T12:44:33ZA fast regularisation of a Newtonian vortex equationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:03b5a5ab-e7bd-4568-9052-d1c84a06b78eEnglishSymplectic ElementsEMS Press2022Carrillo, JAGómez-Castro, DVázquez, JLWe consider equations of the form $ u_t = \nabla \cdot ( \gamma(u) \nabla \mathrm{N}(u))$, where $\mathrm{N}$ is the Newtonian potential (inverse of the Laplacian) posed in the whole space $\mathbb R^d$, and $\gamma(u)$ is the mobility. For linear mobility, $\gamma(u)=u$, the equation and some variations have been proposed as a model for superconductivity or superfluidity. In that case the theory leads to uniqueness of bounded weak solutions having the property of compact space support, and in particular there is a special solution in the form of a disk vortex of constant intensity in space $u=c_1t^{-1}$ supported in a ball that spreads in time like $c_2t^{1/d}$, thus showing a discontinuous leading front. In this paper we propose the model with sublinear mobility $\gamma(u)=u^\alpha$, with $0<\alpha<1$, and prove that nonnegative solutions recover positivity everywhere, and moreover display a fat tail at infinity. The model acts in many ways as a regularization of the previous one. In particular, we find that the equivalent of the previous vortex is an explicit self-similar solution decaying in time like $u=O(t^{-1/\alpha})$ with a space tail with size $u=O(|x|^{- d/(1-\alpha)})$. We restrict the analysis to radial solutions and construct solutions by the method of characteristics. We introduce the mass function, which solves an unusual variation of Burger's equation, and plays an important role in the analysis. We show well-posedness in the sense of viscosity solutions. We also construct numerical finite-difference convergent schemes. |
| spellingShingle | Carrillo, JA Gómez-Castro, D Vázquez, JL A fast regularisation of a Newtonian vortex equation |
| title | A fast regularisation of a Newtonian vortex equation |
| title_full | A fast regularisation of a Newtonian vortex equation |
| title_fullStr | A fast regularisation of a Newtonian vortex equation |
| title_full_unstemmed | A fast regularisation of a Newtonian vortex equation |
| title_short | A fast regularisation of a Newtonian vortex equation |
| title_sort | fast regularisation of a newtonian vortex equation |
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