An Introduction to the Mathematical Theory of the Navier-Stokes Equations

1. Front Matter

   Pages i-xiii

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2. Steady-State Solutions of the Navier-Stokes Equations: Statement of the Problem and Open Questions

   • Giovanni P. Galdi

   Pages 1-16

3. Basic Function Spaces and Related Inequalities

   • Giovanni P. Galdi

   Pages 17-99

4. The Function Spaces of Hydrodynamics

   • Giovanni P. Galdi

   Pages 100-181

5. Steady Stokes Flow in Bounded Domains

   • Giovanni P. Galdi

   Pages 182-243

6. Steady Stokes Flow in Exterior Domains

   • Giovanni P. Galdi

   Pages 244-303

7. Steady Stokes Flow in Domains with Unbounded Boundaries

   • Giovanni P. Galdi

   Pages 304-355

8. Steady Oseen Flow in Exterior Domains

   • Giovanni P. Galdi

   Pages 356-430

9. Back Matter

   Pages 431-465

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Undoubtedly, 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.

• Navier-Stokes equation
• dynamics
• equation
• fluid mechanics
• function

• Istituto di Ingegneria, Università di Ferrara, Ferrara, Italy

  Giovanni P. Galdi

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