Bøker av Alexander G. Ramm

This book revises and expands upon the prior edition, The Navier-Stokes Problem. The focus of this book is to provide a mathematical analysis of the Navier-Stokes Problem (NSP) in R^3 without boundaries. Before delving into analysis, the author begins by explaining the background and history of the Navier-Stokes Problem. This edition includes new analysis and an a priori estimate of the solution. The estimate proves the contradictory nature of the Navier-Stokes Problem. The author reaches the conclusion that the solution to the NSP with smooth and rapidly decaying data cannot exist for all positive times. By proving the NSP paradox, this book provides a solution to the millennium problem concerning the Navier-Stokes Equations and shows that they are physically and mathematically contradictive.

This book revises and expands upon the prior edition, The Navier-Stokes Problem. The focus of this book is to provide a mathematical analysis of the Navier-Stokes Problem (NSP) in R^3 without boundaries. Before delving into analysis, the author begins by explaining the background and history of the Navier-Stokes Problem. This edition includes new analysis and an a priori estimate of the solution. The estimate proves the contradictory nature of the Navier-Stokes Problem. The author reaches the conclusion that the solution to the NSP with smooth and rapidly decaying data cannot exist for all positive times. By proving the NSP paradox, this book provides a solution to the millennium problem concerning the Navier-Stokes Equations and shows that they are physically and mathematically contradictive.

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 ¿+ (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 ¿¿¿¿ ¿ ¿+, 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(¿3) × C(¿+) is proved, ¿¿¿¿21(¿3) is the Sobolev space, ¿+ = [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.

This book gives a necessary and sufficient condition in terms of the scattering amplitude for a scatterer to be spherically symmetric. By a scatterer we mean a potential or an obstacle. It also gives necessary and sufficient conditions for a domain to be a ball if an overdetermined boundary problem for the Helmholtz equation in this domain is solvable. This includes a proof of Schiffer's conjecture, the solution to the Pompeiu problem, and other symmetry problems for partial differential equations. It goes on to study some other symmetry problems related to the potential theory. Among these is the problem of "e;invisible obstacles."e; In Chapter 5, it provides a solution to the Navier‒Stokes problem in ℝ³. The author proves that this problem has a unique global solution if the data are smooth and decaying sufficiently fast. A new a priori estimate of the solution to the Navier‒Stokes problem is also included. Finally, it delivers a solution to inverse problem of the potential theory without the standard assumptions about star-shapeness of the homogeneous bodies.

The inverse obstacle scattering problem consists of finding the unknown surface of a body (obstacle) from the scattering ,,,,(,,,,;,,,,;,,,,), where ,,,,(,,,,;,,,,;,,,,) is the scattering amplitude, ,,,,;,,,, ,,,, ,,,,2 is the direction of the scattered, incident wave, respectively, ,,,,2 is the unit sphere in the and k > 0 is the modulus of the wave vector. The scattering data is called non-over-determined if its dimensionality is the same as the one of the unknown object. By the dimensionality one understands the minimal number of variables of a function describing the data or an object. In an inverse obstacle scattering problem this number is 2, and an example of non-over-determined data is ,,,,(,,,,) := ,,,,(,,,,;,,,,,, By sub-index 0 a fixed value of a variable is denoted.</p>It is proved in this book that the data ,,,,(,,,,), known for all ,,,, in an open subset of ,,,, determines uniquely the surface ,,,, and the boundary condition on ,,,,. This condition can be the Dirichlet, or the Neumann, or the impedance type.</p>The above uniqueness theorem is of principal importance because the non-over-determined data are the minimal data determining uniquely the unknown ,,,,. There were no such results in the literature, therefore the need for this book arose. This book contains a self-contained proof of the existence and uniqueness of the scattering solution for rough surfaces.</p>

Radon Transform and Local Tomography presents new theories and computational methods that cannot be found in any other book. New material, aimed at solving important problems in tomographic imaging and image processing in general, as well as detailed descriptions of the new algorithms and the results of their testing, are expertly covered. The theory described in this book solves the important problem of finding discontinuities of function from its tomographic data. A detailed theoretical analysis and three different solutions to this problem are given, as well as algorithms for practical solutions and examples of applications for both simulated and real-life data.

Radon Transform and Local Tomography presents new theories and computational methods that cannot be found in any other book. New material, aimed at solving important problems in tomographic imaging and image processing in general, as well as detailed descriptions of the new algorithms and the results of their testing, are expertly covered. The theory described in this book solves the important problem of finding discontinuities of function from its tomographic data. A detailed theoretical analysis and three different solutions to this problem are given, as well as algorithms for practical solutions and examples of applications for both simulated and real-life data.