Read e-book online An Introduction to the Mathematical Theory of the PDF

By G.P. Galdi (auth.)

ISBN-10: 0387096191

ISBN-13: 9780387096193

The booklet presents a complete, special and self-contained therapy of the elemental mathematical houses of boundary-value difficulties relating to the Navier-Stokes equations. those homes contain life, area of expertise and regularity of recommendations in bounded in addition to unbounded domain names. each time the area is unbounded, the asymptotic habit of recommendations is additionally investigated.

This booklet is the recent version of the unique quantity publication, below a similar identify, released in 1994.

In this re-creation, the 2 volumes have merged into one and extra chapters on regular generalized oseen stream in external domain names and regular Navier–Stokes circulation in three-d external domain names were additional. many of the proofs given within the past version have been additionally updated.

An introductory first bankruptcy describes all proper questions handled within the ebook and lists and motivates a few major and nonetheless open questions. it's written in an expository sort as a way to be available additionally to non-specialists. each one bankruptcy is preceded by means of a considerable, initial dialogue of the issues handled, in addition to their motivation and the tactic used to resolve them. additionally, each one bankruptcy ends with a piece devoted to replacement methods and approaches, in addition to ancient notes.

The publication comprises greater than four hundred stimulating workouts, at diversified degrees of hassle, that would support the junior researcher and the graduate scholar to progressively develop into accustomed with the topic. ultimately, the booklet is endowed with an enormous bibliography that incorporates greater than 500 goods. every one merchandise brings a connection with the part of the ebook the place it truly is mentioned.

The e-book should be important to researchers and graduate scholars in arithmetic specifically mathematical fluid mechanics and differential equations.

Review of First version, First Volume:

“The emphasis of this e-book is on an advent to the mathematical idea of the desk bound Navier-Stokes equations. it truly is written within the form of a textbook and is largely self-contained. the issues are awarded essentially and in an obtainable demeanour. each bankruptcy starts with a great introductory dialogue of the issues thought of, and ends with fascinating notes on various ways constructed within the literature. additional, stimulating routines are proposed. (Mathematical reports, 1995)

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Additional resources for An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems

Example text

2), and b∗ is a prescribed vector field. 1), and v is totally prescribed (no slip), while if 1/β → 0, only v · n is prescribed, and we lose information on the tangential component v τ (pure slip). However, if β = 0 and finite, (∗) allows for v τ to be nonzero, by an amount that depends on the magnitude of the tangential stress at the boundary (partial slip). The Navier condition (∗), with 1/β = 0 or → 0, has been employed in a wide range of problems. , Xiao & Xin 2007, Beir˜ ao da Veiga 2010). Concerning the use of (∗) in steady-state studies, after the pioneering work ˇcadilov (1973), where pure slip boundary conditions are of Solonnikov and Sˇ used along with a linearized system of equations (Stokes equations), in the last few years there has been considerably increasing interest.

Besides the papers of Beir˜ao da Veiga (2004, 2005), which generalize and simplify the proof of the ˇcadilov, we refer the interested reader, for example, results of Solonnikov & Sˇ to Ebemeyer & Frehse (2001) for flow in bounded domains, Mucha (2003), Konieczny (2006), and Beir˜ ao da Veiga (2006) for flow in infinite channels and pipes, to Konieczny (2009) for flow in exterior domains, and to the literature cited therein. 2). 3). 1), with the objective of explaining the difference between the discharges in glass and copper tubes, as experimentally observed by Girard (1816).

1) i=0 where Ω0 is a smooth bounded subset of Ω, while Ωi , i = 1, 2, are disjoint regions that, in possibly different coordinate systems (depending on Ωi , i = 1, 2), reduce to straight cylinders (strips, for n = 2), that is, Ωi = {x ∈ Rn : xn > 0, x ≡ (x1 , . . 2) n−1 with Σi bounded and simply connected regions in R . Denoting by Σ any bounded intersection of Ω with a plane, which in Ωi reduces to Σi , and by n a unit vector orthogonal to Σ, oriented from Ω1 toward Ω2 (say) owing to the incompressibility of the liquid and assuming that v vanishes at the boundary, we at once deduce that the flux Φ of v through Σ is a constant: Φ≡ Σ v · n = const.

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An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems by G.P. Galdi (auth.)


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