The author approaches an old classic problem - the existence of solutions of Navier-Stokes equations. The main objective is to model and derive of equation of continuity, Euler equation of fluid motion, energy flux equation, Navier-Stokes equations from the observer point of view and solve classic problem for this interpretation of fluid motion laws.

If we have a piece of metal or a volume of liquid, the idea impresses itself upon us that it is divisible without limit, that any part of it, however small, would again have the same properties. But, wherever the methods of research in the physics of matter were refined sufficiently, limits to divisibility were reached that are not due to the inadequacy of our experiments but to the nature of the subject matter.

Observability in mathematics were developed by the author based on denial of infinity idea. He introduces observers into arithmetic, and arithmetic becomes dependent on observers. And after that the basic mathematical parts also become dependent on observers. This approach permits to reconsider the fluid motion laws, analyze them and get solutions of classic problems.

**Table of Contents**

1. Introduction.

2. Observability and Arithmetic.

3. Observability and Vector Algebra.

4. Observability and Mathematical Analysis (Calculus).

5. Classic Fluid Mechanics equations and Observability.

6. Observability and Thermodynamical equations.

7. Observability and equation of continuity.

8. Observability and Euler equation of motion of the fluid.

9. Observability and energy flux and moment flux equations.

10. Observability and incompressible fluids.

11. Observability and Navier-Stokes equations.

12. Observability and Relativistic Fluid Mechanics.

13. Appendix: Review of publications of the Mathematics with Observers.

14. Glossary.

Bibliography

Index

**Biography**

Boris Khots, DrSci, lives in Iowa, USA, Independent Researcher. Alma Mater - Moscow State Lomonosov University, Department of Mathematics and Mechanics (mech-math). Creator of Observer's Mathematics. Participant of more than 30 Mathematical international congresses, conferences. In particular, participated with presentation at International Congresses of Mathematicians on 1998 (Germany), 2002 (China), 2006 (Spain), 2010 (India), 2014 (South Korea). More than 150 mathematical books and papers.