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This article is cited in 4 scientific papers (total in 4 papers)
MATHEMATICS
On totally global solvability of controlled second kind operator equation
A. V. Chernovab a Nizhny
Novgorod State University, pr. Gagarina, 23, Nizhny Novgorod, 603950, Russia
b Nizhny Novgorod State Technical University, ul. Minina, 24, Nizhny Novgorod, 603950, Russia
Abstract:
We consider the nonlinear evolutionary operator equation
of the second kind as follows
$\varphi=\mathcal{F}\bigl[f[u]\varphi\bigr]$,
$\varphi\in W[0;T]\subset L_q\bigl([0;T];X\bigr)$,
with Volterra type operators
$\mathcal{F}\colon L_p\bigl([0;\tau];Y\bigr)\to W[0;T]$,
$f[u]$:
$W[0;T]\to L_p\bigl([0;T];Y\bigr)$
of the general form,
a control $u\in\mathcal{D}$
and arbitrary Banach spaces $X$, $Y$.
For this equation we prove
theorems on solution uniqueness and
sufficient conditions for
totally (with respect to set $\mathcal{D}$)
global solvability.
Under natural hypotheses associated with
pointwise in $t\in[0;T]$ estimates
the conclusion on univalent totally global solvability
is made provided global solvability for
a comparison system which is some system of
functional integral equations (it could be
replaced by a system of equations of analogous type,
and in some cases, of ordinary differential equations)
with respect to unknown functions
$[0;T]\to\mathbb{R}$.
As an example we establish sufficient conditions of
univalent totally global solvability for a controlled
nonlinear nonstationary Navier–Stokes system.
Keywords:
nonlinear evolutionary operator equation of the second kind, totally global solvability, Navier–Stokes system.
Received: 23.08.2019
Citation:
A. V. Chernov, “On totally global solvability of controlled second kind operator equation”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 30:1 (2020), 92–111
Linking options:
https://www.mathnet.ru/eng/vuu712 https://www.mathnet.ru/eng/vuu/v30/i1/p92
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