Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains

Solvability of <i>p</i>-Laplacian Parabolic Equations with Constraints Coupled with Navier-Stokes Equations in 3D Domains by Using Largeness of <i>p</i>

“…Let us recall the previous works [9], [22] and [24] in 2-dimensional domains. In [9] and [22] it is shown that there exists a unique solution ðy; vÞ to the problem (P) with p ¼ 2 such that [24] it is proved that the problem (P) with p > 2 admits a unique solution in the classes (C2) and Indeed, in the case p ¼ 2 as in [9] and [22], noting that and hence (1.1) holds.…”

“…In [9] and [22] it is shown that there exists a unique solution ðy; vÞ to the problem (P) with p ¼ 2 such that [24] it is proved that the problem (P) with p > 2 admits a unique solution in the classes (C2) and Indeed, in the case p ¼ 2 as in [9] and [22], noting that and hence (1.1) holds. The other of the two important parts is to introduce the two mappings, which are for example the two solution operators of the heat equation with fixed velocity and the Navier-Stokes equation with fixed temperature, and to construct solutions to (P) by combining the two mappings.…”

“…The other of the two important parts is to introduce the two mappings, which are for example the two solution operators of the heat equation with fixed velocity and the Navier-Stokes equation with fixed temperature, and to construct solutions to (P) by combining the two mappings. Indeed in the both case p ¼ 2 and p > 2, they apply the Schauder fixed point theorem (as in [9]) or the contraction mapping principle (as in [22] and [24]) to the composition of the two mappings mentioned above.…”

“…Fukao-Kubo [9] and Sobajima-Tsuzuki-Yokota [22] established the solvability of the problem (P) with the di¤usion term D p y altered into Dy in 2-dimensional domains. Tsuzuki [24] also solved the problem (P) with the di¤usion term D p y in 2-dimensional domains. Other problems associated with the Boussinesq system are dealt with in many papers (see e.g., Morimoto [15], Kubo [13], Fukao-Kenmochi [6] and Fukao-Kubo [8]).…”

“…However the works adopting the operator theory with the solvability of (P), namely [9], [22] and [24], are done only in 2-dimensional domains. It should be appreciated that we discuss the solvability in 3-dimensional domains even if we consider an artificial setting with p-Laplacian.…”