On the Duhamel’s Solutions to the Null Equations of Incompressible Fluids

Abstract

In this paper, we have solved the Diophantine equation ቄ൫ ൯ ቅ + = ࢠ and ቄ൫ ൯ ቅ + = ࢠ where ∈ ା and p is an odd prime. Also, we have discussed the generalization of ሺ ሻ + = ࢠ to ቄ൫ ൯ ቅ + = ࢠ , where ∈ ା , q is any prime number and p is an odd prime number. Some solutions of these Diophantine equations have been obtained.

The Mathematical Description of Homogeneous Turbulence for Incompressible Fluids

“…1) The turbulence modelled by of the Navier-Stokes's momentum equations means that the actual realizable Navier Stokes momentum equation solution is of a probabilistic formulation [3,6].…”

“…Even if we could solve the nonlinear Lotka-Volterra system of "differential" equations it would yield an integral equation involving spatial operators as "parameters". Currently, the Navier-Stokes's momentum equations full nontrivial solution remains unknown unless a simplifying assumption is made, for example zero initial conditions [3] which provides the linearizing condition, ∑ ‫ݍ‬ ‫ݑ‬ ‫ݑ‬ = 0. A specialized solution of the Navier-Stokes with zero initial conditions or null incompressible Navier Stokes equations given the input pressure and external forces are known has been solved in the reference article [3] shows the nature of the Greens function kernel (݁ ௧ఔ − 1ሻ is essentially related to the normal Gaussian distribution function.…”

“…Currently, the Navier-Stokes's momentum equations full nontrivial solution remains unknown unless a simplifying assumption is made, for example zero initial conditions [3] which provides the linearizing condition, ∑ ‫ݍ‬ ‫ݑ‬ ‫ݑ‬ = 0. A specialized solution of the Navier-Stokes with zero initial conditions or null incompressible Navier Stokes equations given the input pressure and external forces are known has been solved in the reference article [3] shows the nature of the Greens function kernel (݁ ௧ఔ − 1ሻ is essentially related to the normal Gaussian distribution function. The exponential Laplacian operator, ݁ ௧ఔ , operating on some function is equivalent to the solution of the heat equation with the initial condition being the function, according to Professor Terence Tao in reference [5] in page 39.…”

“…The exponential Laplacian operator, ݁ ௧ఔ , operating on some function is equivalent to the solution of the heat equation with the initial condition being the function, according to Professor Terence Tao in reference [5] in page 39. By integrating by parts in the time integral specified in Theorem 1 of reference [3] and using the relationship provided by [5] to obtain the null velocity field in terms of Gaussian distribution expectations as shown below provided the null velocity solution, ܷ , satisfies ∑ ܷ…”

“…Where ߰ ൫߬, ‫ݔ‬ ሺ߬, ‫ݔ‬ ሻ൯ is set to zero in Eq. 7 of reference [3] and ܹ ൫߬, ‫ݔ‬ ሺ߬, ‫ݔ‬ ሻ൯ is given as follows:…”