Solutions to the stochastic thin-film equation for the range of mobility exponents $n\in (2,3)$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918005875671040 |
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| author | Sauerbrey, Max |
| author_facet | Sauerbrey, Max |
| contents | Recently, many existence results for the stochastic thin-film equation were established in the case of a quadratic mobility exponent $n=2$, in which the noise term $\partial_x(u^\frac{n}{2}\mathcal{W})$ becomes linear. In the case of a non-quadratic mobility exponent, results are only available in the situation that $n\ge \frac{8}{3}$ leaving the interval of mobility exponents $n\in (2,\frac{8}{3})$ untreated. In this article we resolve the current gap in the literature by presenting a proof, which works under the assumption $n\in (2,3)$, i.e., the regime of weak slippage. The key idea is to use that the $\log$-entropy dissipation coincides with the energy production due to the noise. To realize this idea, we approximate the stochastic thin-film equation by stochastic thin-film equations with inhomogeneous mobility functions, which behave like a higher power near $0$. As a consequence the approximate solutions are non-negative, which is vital to use the $\log$-entropy estimate. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_02765 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Solutions to the stochastic thin-film equation for the range of mobility exponents $n\in (2,3)$ Sauerbrey, Max Probability Analysis of PDEs 35R60, 76A20 Recently, many existence results for the stochastic thin-film equation were established in the case of a quadratic mobility exponent $n=2$, in which the noise term $\partial_x(u^\frac{n}{2}\mathcal{W})$ becomes linear. In the case of a non-quadratic mobility exponent, results are only available in the situation that $n\ge \frac{8}{3}$ leaving the interval of mobility exponents $n\in (2,\frac{8}{3})$ untreated. In this article we resolve the current gap in the literature by presenting a proof, which works under the assumption $n\in (2,3)$, i.e., the regime of weak slippage. The key idea is to use that the $\log$-entropy dissipation coincides with the energy production due to the noise. To realize this idea, we approximate the stochastic thin-film equation by stochastic thin-film equations with inhomogeneous mobility functions, which behave like a higher power near $0$. As a consequence the approximate solutions are non-negative, which is vital to use the $\log$-entropy estimate. |
| title | Solutions to the stochastic thin-film equation for the range of mobility exponents $n\in (2,3)$ |
| topic | Probability Analysis of PDEs 35R60, 76A20 |
| url | https://arxiv.org/abs/2310.02765 |