The Navier-Stokes Equations

1. Front Matter

   Pages i-x

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2. Introduction

   • Hermann Sohr

   Pages 1-42

3. Preliminary Results

   • Hermann Sohr

   Pages 43-106

4. The Stationary Navier-Stokes Equations

   • Hermann Sohr

   Pages 107-184

5. The Linearized Nonstationary Theory

   • Hermann Sohr

   Pages 185-260

6. The Full Nonlinear Navier-Stokes Equations

   • Hermann Sohr

   Pages 261-353

7. Back Matter

   Pages 355-367

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The primary objective of this monograph is to develop an elementary and se- containedapproachtothemathematicaltheoryofaviscousincompressible?uid n in a domain ? of the Euclidean spaceR , described by the equations of Navier- Stokes. The book is mainly directed to students familiar with basic functional analytic tools in Hilbert and Banach spaces. However, for readers’ convenience, in the ?rst two chapters we collect, without proof some fundamental properties of Sobolev spaces, distributions, operators, etc. Another important objective is to formulate the theory for a completely general domain ?. In particular, the theory applies to arbitrary unbounded, non-smooth domains. For this reason, in the nonlinear case, we have to restrict ourselves to space dimensions n=2,3 that are also most signi?cant from the physical point of view. For mathematical generality, we will develop the l- earized theory for all n? 2. Although the functional-analytic approach developed here is, in principle, known to specialists, its systematic treatment is not available, and even the diverseaspectsavailablearespreadoutintheliterature.However,theliterature is very wide, and I did not even try to include a full list of related papers, also because this could be confusing for the student. In this regard, I would like to apologize for not quoting all the works that, directly or indirectly, have inspired this monograph.

• Navier-Stokes equation
• Sobolev space
• distribution
• fluid mechanics
• stationary and nonstationary fluids
• viscous incompressible fluids
• partial differential equations

From the reviews:

“The book is well written and not unnecessarily wordy. There is an up-to-date bibliography and a nice index. … a mathematician who wishes to know what the important issues concerning eq. (1) are and what has been achieved, would find this an excellent source. Equally, a mathematically-minded student, with a good grounding in analysis and who has decided to work in this area, or the teacher who wants to teach a course on this material would find this a valuable text.” (P. N. Shankar,Current Science, Vol. 85 (2), July, 2003)

• , Department of Mathematics, University Paderborn, Paderborn, Germany

  Hermann Sohr

Hermann Sohr is professor of mathematics, especially mathematical fluid mechanics, at the University Paderborn, Germany.

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