An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Volume I: Linearised Steady ProblemsUndoubtedly, the Navier-Stokes equations are of basic importance within the context of modern theory of partial differential equations. Although the range of their applicability to concrete problems has now been clearly recognised to be limited, as my dear friend and bright colleague K.R. Ra jagopal has showed me by several examples during the past six years, the mathematical questions that remain open are of such a fascinating and challenging nature that analysts and applied mathematicians cannot help being attracted by them and trying to contribute to their resolution. Thus, it is not a coincidence that over the past ten years more than seventy sig nificant research papers have appeared concerning the well-posedness of boundary and initial-boundary value problems. In this monograph I shall perform a systematic and up-to-date investiga tion of the fundamental properties of the Navier-Stokes equations, including existence, uniqueness, and regularity of solutions and, whenever the region of flow is unbounded, of their spatial asymptotic behavior. I shall omit other relevant topics like boundary layer theory, stability, bifurcation, de tailed analysis of the behavior for large times, and free-boundary problems, which are to be considered "advanced" ones. In this sense the present work should be regarded as "introductory" to the matter. |
Contents
Preface to the First Revised Edition | 1 |
Basic Function Spaces and Related Inequalities | 17 |
17 | 38 |
100 | 116 |
Steady Stokes Flow in Bounded Domains | 182 |
Domain The Greens Tensor | 227 |
Steady Stokes Flow in Exterior Domains | 244 |
Large Distances and Related Results | 254 |
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An introduction to the mathematical theory of the Navier-Stokes equations Giovanni P. Galdi No preview available - 2014 |
Common terms and phrases
analogous approximation arbitrary Assume assumptions asymptotic Banach space Bogovskii boundary bounded domain c₁ Chapter Co(N compact compact support condition cone condition consider constant convergence corresponding D¹¹ª(N deduce defined denote derive differentiable Dirichlet integral estimate Exercise existence exterior domain finite flow fluid flux function fundamental solution Galdi generalised solution given H₁(N Hölder inequality identity implies independent infinity integral Lª(N large distances Lebesgue spaces Lemma linear Math Moreover Navier-Stokes Equations norm Notes obtain Oseen particular Pileckas properties prove q-generalised r₁ Remark right-hand side satisfies Section sequence Simader smooth smooth functions Sobolev inequality Sobolev spaces Sohr solenoidal Solenoidal Vector Fields solvability Stokes problem Stokes system suitable tensor Theorem 2.1 unbounded uniqueness v₁ validity vanishing vector field velocity field verifies weak solution weakly Wm,q zero მი