Local-in-time strong solvability of Navier--Stokes type variational inequalities by Rothe's method
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918288525623296 |
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| author | Kashiwabara, Takahito |
| author_facet | Kashiwabara, Takahito |
| contents | We consider parabolic variational inequalities in a Hilbert space $V$, which have a non-monotone nonlinearity of Navier--Stokes type represented by a bilinear operator $B: V \times V \to V'$ and a monotone type nonlinearity described by a convex, proper, and lower-semicontinuous functional $φ: V \to (-\infty, +\infty]$. Existence and uniqueness of a local-in-time strong solution in a maximal-$L^2$-regularity class and in a Kiselev--Ladyzhenskaya class are proved by discretization in time (also known as Rothe's method), provided that a corresponding stationary Stokes problem admits a regularity structure better than $V$ (which is typically $H^2$-regularity in case of the Navier--Stokes equations). Since we do not assume the cancelation property $\left< B(u, v), v \right> = 0$, in applications we may allow for broader boundary conditions than those treated by the existing literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09190 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local-in-time strong solvability of Navier--Stokes type variational inequalities by Rothe's method Kashiwabara, Takahito Analysis of PDEs We consider parabolic variational inequalities in a Hilbert space $V$, which have a non-monotone nonlinearity of Navier--Stokes type represented by a bilinear operator $B: V \times V \to V'$ and a monotone type nonlinearity described by a convex, proper, and lower-semicontinuous functional $φ: V \to (-\infty, +\infty]$. Existence and uniqueness of a local-in-time strong solution in a maximal-$L^2$-regularity class and in a Kiselev--Ladyzhenskaya class are proved by discretization in time (also known as Rothe's method), provided that a corresponding stationary Stokes problem admits a regularity structure better than $V$ (which is typically $H^2$-regularity in case of the Navier--Stokes equations). Since we do not assume the cancelation property $\left< B(u, v), v \right> = 0$, in applications we may allow for broader boundary conditions than those treated by the existing literature. |
| title | Local-in-time strong solvability of Navier--Stokes type variational inequalities by Rothe's method |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.09190 |