The Large Flux Problem To The Navier-stokes Equations: Global Strong Solutions In Cylindrical Domains (advances In Mathematical Fluid Mechanics)

This monograph considers the motion of incompressible fluids described by the Navier-Stokes equations with large inflow and outflow, and proves the existence of global regular solutions without any restrictions on the magnitude of the initial velocity, the external force, or the flux. To accomplish this, some assumptions are necessary: The flux is close to homogeneous, and the initial velocity and the external force do not change too much along the axis of the cylinder. This is achieved by utilizing a sophisticated method of deriving energy type estimates for weak solutions and global estimates for regular solutions—an approach that is wholly unique within the existing literature on the Navier-Stokes equations. To demonstrate these results, three main steps are followed: first, the existence of weak solutions is shown; next, the conditions guaranteeing the regularity of weak solutions are presented; and, lastly, global regular solutions are proven. This volume is ideal for mathematicians whose work involves the Navier-Stokes equations, and, more broadly, researchers studying fluid mechanics. Front Matter ....Pages i-vi Introduction (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 1-9 Notation and Auxiliary Results (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 11-30 Energy Estimate: Global Weak Solutions (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 31-57 Local Estimates for Regular Solutions (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 59-82 Global Estimates for Solutions to Problem on (v, p) (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 83-93 Global Estimates for Solutions to Problem on (h, q) (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 95-97 Estimates for ht (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 99-105 Auxiliary Results: Estimates for (v, p) (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 107-115 Auxiliary Results: Estimates for (h, q) (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 117-128 The Neumann Problem (3.6) in L2-Weighted Spaces (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 129-141 The Neumann Problem (3.6) in Lp-Weighted Spaces (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 143-155 Existence of Solutions (v, p) and (h, q) (Joanna Rencławowicz, Wojciech M. Zajączkowski)....Pages 157-168 Back Matter ....Pages 169-179 Contents 6 1 Introduction 8 2 Notation and Auxiliary Results 17 2.1 Spaces and Basic Notation 17 2.2 Auxiliary Problems, Results, and Green Functions 23 2.3 Interpolation, Imbeddings, Trace Theorems, and the Korn Inequality 30 3 Energy Estimate: Global Weak Solutions 37 3.1 Weak Solutions 40 3.2 Estimates for vt 53 4 Local Estimates for Regular Solutions 64 4.1 A Priori Estimates for Function h=v,x3 69 4.2 A Priori Estimates for Vorticity Component χ 77 4.3 Relating v and h: rot–div System 80 5 Global Estimates for Solutions to Problem on (v,p) 88 6 Global Estimates for Solutions to Problem on (h,q) 99 7 Estimates for ht 102 8 Auxiliary Results: Estimates for (v,p) 109 9 Auxiliary Results: Estimates for (h,q) 118 10 The Neumann Problem (3.6) in L2-Weighted Spaces 130 11 The Neumann Problem (3.6) in Lp-Weighted Spaces 143 12 Existence of Solutions (v,p) and (h,q) 156 12.1 Existence of Weak Solutions 156 12.2 Existence of Regular Solutions 163 References 168 Notation Index 172 Name Index 175 Subject Index 176