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The Navier-Stokes Problem [electronic resource] / by Alexander G. Ramm.

By: Ramm, Alexander G [author.].
Contributor(s): SpringerLink (Online service).
Material type: materialTypeLabelBookSeries: Synthesis Lectures on Mathematics & Statistics: Publisher: Cham : Springer International Publishing : Imprint: Springer, 2021Edition: 1st ed. 2021.Description: XV, 61 p. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9783031024313.Subject(s): Mathematics | Statistics  | Engineering mathematics | Mathematics | Statistics | Engineering MathematicsAdditional physical formats: Printed edition:: No title; Printed edition:: No title; Printed edition:: No titleDDC classification: 510 Online resources: Click here to access online
Contents:
Preface -- Introduction -- Brief History of the Navier-Stokes Problem -- Statement of the Navier-Stokes Problem -- Theory of Some Hyper-Singular Integral Equations -- A Priori Estimates of the Solution to the NSP -- Uniqueness of the Solution to the NSP -- The Paradox and its Consequences -- Logical Analysis of Our Proof -- Appendix 1 - Theory of Distributions and Hyper-Singular Integrals -- Appendix 2 - Gamma and Beta Functions -- Appendix 3 - The Laplace Transform -- Bibliography -- Author's Biography.
In: Springer Nature eBookSummary: The main result of this book is a proof of the contradictory nature of the Navier‒Stokes problem (NSP). It is proved that the NSP is physically wrong, and the solution to the NSP does not exist on ℝ+ (except for the case when the initial velocity and the exterior force are both equal to zero; in this case, the solution ����(����, ����) to the NSP exists for all ���� ≥ 0 and ����(����, ����) = 0). It is shown that if the initial data ����0(����) ≢ 0, ����(����,����) = 0 and the solution to the NSP exists for all ���� ϵ ℝ+, then ����0(����) := ����(����, 0) = 0. This Paradox proves that the NSP is physically incorrect and mathematically unsolvable, in general. Uniqueness of the solution to the NSP in the space ����21(ℝ3) × C(ℝ+) is proved, ����21(ℝ3) is the Sobolev space, ℝ+ = [0, ∞). Theory of integral equations and inequalities with hyper-singular kernels is developed. The NSP is reduced to an integral inequality with a hyper-singular kernel.
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Preface -- Introduction -- Brief History of the Navier-Stokes Problem -- Statement of the Navier-Stokes Problem -- Theory of Some Hyper-Singular Integral Equations -- A Priori Estimates of the Solution to the NSP -- Uniqueness of the Solution to the NSP -- The Paradox and its Consequences -- Logical Analysis of Our Proof -- Appendix 1 - Theory of Distributions and Hyper-Singular Integrals -- Appendix 2 - Gamma and Beta Functions -- Appendix 3 - The Laplace Transform -- Bibliography -- Author's Biography.

The main result of this book is a proof of the contradictory nature of the Navier‒Stokes problem (NSP). It is proved that the NSP is physically wrong, and the solution to the NSP does not exist on ℝ+ (except for the case when the initial velocity and the exterior force are both equal to zero; in this case, the solution ����(����, ����) to the NSP exists for all ���� ≥ 0 and ����(����, ����) = 0). It is shown that if the initial data ����0(����) ≢ 0, ����(����,����) = 0 and the solution to the NSP exists for all ���� ϵ ℝ+, then ����0(����) := ����(����, 0) = 0. This Paradox proves that the NSP is physically incorrect and mathematically unsolvable, in general. Uniqueness of the solution to the NSP in the space ����21(ℝ3) × C(ℝ+) is proved, ����21(ℝ3) is the Sobolev space, ℝ+ = [0, ∞). Theory of integral equations and inequalities with hyper-singular kernels is developed. The NSP is reduced to an integral inequality with a hyper-singular kernel.

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