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Конференция международных математических центров мирового уровня
13 августа 2021 г. 14:30–15:20, Комплексный анализ, г. Сочи
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An open mapping theorem for nonlinear operator equations associated with the Dolbeault complex
А. А. Шлапунов Научно-технологический университет "Сириус", г. Сочи
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Аннотация:
Let $ \{\overline \partial^{p,q},\Lambda^{p,q}\} $, $n\geq 1$, be the Dolbeault complex over
the complex space ${\mathbb C}^n $ where $\Lambda^{p,q}$ be the trivial vector bundle
of the exterior differential forms of bidegree $(p,q)$. Denote by $(\overline \partial^{p,q})^*$ the formal adjoint differential operator for
the Dolbeault differential $\overline \partial^{p,q}$ related to the standard Hermitian
structure of the Lebesgue space $L^2 _{\Lambda^{p,q}}({\mathbb C}^n)$ consisting
of the forms of the corresponding bi-degree. Then
the Laplacians $\Delta^{p,q} = (\overline \partial^{p,q})^* \overline \partial^{p,q}
+ \overline \partial^{p,q-1} (\overline \partial^{p,q-1})^*$ of the
complex are
strongly elliptic and, with any fixed number $\mu>0$, the
operators $\partial_t + \mu \Delta^{p,q} $, acting
on the induced bundle $\Lambda^{p,q} (t)$ consisting of differential forms with
coefficients, depending on the variable $t \in [0,T]$, $T>0$, as a parameter, are parabolic.
We consider the following initial problem: given section $ f $ of the induced bundle
$ \Lambda^{p,q} (t) $ and $(p,q)$-form $ u_0 $, find sections
$ u,v $ of the bundles $ \Lambda^{p,q} (t) $ and
$ \Lambda^{p,q-1} (t) $, respectively, such that
\begin{equation} \label{eq.NS.Dolbeault}
\begin{cases}
\partial_t u + \mu \Delta^{p,q} u + N^{p,q} (u)
+ \overline \partial^{p,q-1} v = f& \text{in } {\mathbb C}^n \times (0,T),\\
(\overline \partial^{p,q-1})^* u =0 , \, (\overline \partial^{p,q-2})^* v =0 ,
& \text{in } {\mathbb C}^n \times [0,T],\\
u(x,0) = u_0& \text{in } {\mathbb C}^n,
\end{cases}
\end{equation}
where the non-linearity
$
N^{p,q} u = M_{p,q,1} ( \overline \partial^{p,q}u, u) + \overline \partial^{p,q-1}
M_{p,q,2}(u, u)$ is generated by a rather wide class of linear bi-differential operators
$M_{p,q,j}$ of zero order,
in specially constructed Bochner-Sobolev type spaces over ${\mathbb C}^n$.
Using the standard technique adapted to study Navier-Stokes' type equations
(the energy type estimates, the interpolation Gagliardo-Nirenberg inequalities and
the Faedo-Galerkin method), see, for instance, [1], we
prove that, under reasonable assumptions regarding the nonlinear term,
the Frechét derivative $ \mathcal{A}_{p,q}' $ of
the nonlinear mapping $ \mathcal{A}_{p,q} $,
induced by \eqref{eq.NS.Dolbeault}, is continuously invertible and the mapping
$ \mathcal{A}_{p,q} $ itself is open and injective in chosen spaces.
However for an Existence Theorem related to even weak (distributional)
solutions to \eqref{eq.NS.Dolbeault} one should necessarily assume that the bilinear operators $M_{p,q,1}$
have additional rather restrictive but natural properties.
The thesis is based on the joint results with A.N. Polkovnikov (Siberian Federal University). The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS".
Website:
https://talantiuspeh.webex.com/talantiuspeh-ru/j.php?MTID=mf31d5efe7cc481a97135e79e32db81fe
Список литературы
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J.-L. Lions, “Quelques méthodes de résolution des problèmes aux limites non linéare”, Dunod/Gauthier-Villars, Paris, 1969, 588 pp.
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