Abstract:
In this paper, the Cauchy problem for the stationary and nonstationary nonlocal incompressible abstract Stokes equation is considered. The equation involves a convolution term and an abstract operator in a Banach space $E$. Existence, uniqueness, and coercive estimates are derived in $L^{p}$ spaces. Different classes of Stokes equations can be obtained by choosing the space $E$ and the linear operator $A$, which occur in a wide variety of physical systems. As an application of the obtained results, the existence, uniqueness, and $L^{p}$-maximal regularity properties of solutions to initial value problems for nonlocal degenerate Stokes equations and nonlocal Stokes equations with discontinuous coefficients are established.
Keywords:Stokes system, elliptic equation, parabolic equation, semigroups of operators, boundary value problem, abstract differential equation, maximal $L^{p}$-regularity.
Fulltext:
PDF file (462 kB)
First page:
PDF file