previous overview

Albert Einstein, "we can't solve problems by using the same kind of thinking we used when we created them",

Wolfgang E. Pauli, "all things reach the one who knows how to wait".

This homepage addresses the following three Millennium problems (resp. links to corresponding homepages):

A. The Riemann Hypothesis (RH)               
B. The 3D-Navier-Stokes equations (NSE)
C. The Yang-Mills equations (YME)     

Our proposed solutions are building on or sharing common mathematical solution concepts and tools (e.g. Hilbert scale, Hilbert (resp. Riesz) transform(s), Hilbert-Polya conjecture).

With respect to the RH we note that the Müntz's formula can not be applied to prove the RH based on the Polya criterion. This is due to the divergence of the Müntz formala in the critical stripe and the regularity and asymptotic assumptions to the baseline function. A distributional Hilbert space provides a framework to address the underlying mathematical issue. In same distributional Hilbert space frame the Bagchi reformulation of the Nyman-Beurling RH criterion enables the Hilbert-Polya conjecture. In this context we note that the Zeta function is an integral function of order 1 and an element of the distributional Hilbert space H(-1).

In the following document we provide a high level walkthrough to those topics (latest update : August 24, 2018, update section A: pp. 5-7, section C p. 12, section C2: pp. 19-20, section D pp. 29-30)


Braun K., Three Millennium problems, RH, NSE, YME, OVERVIEW

With respect to the NSE and the YME the proposed mathematical concepts and tools are especially correlated to the names of Plemelj, Stieltjes and Calderón.

The essential estimate for the critical non-linear term of the non-linear, non-stationary 3-D NSE  has been provided by Sobolevskii.

With respect to the YME the proposed mathematical concepts and tools are especially correlated to the names of Schrödinger and Weyl (e.g. in the context of "half-odd integers quantum numbers for the Bose statistics" and "a truly infinitesimal geometry"). It enables an alternative (quantum) ground state energy model embedded in the proposed distributional Hilbert scale frame of this homepage covering all variational physical-mathematical PDE and Pseudo differential equations (e.g. also the Maxwell equations).

The Dirac theory with its underlying concept of a "Dirac function" is omitted and replaced by a distributional Hilbert space (domain) concept. This alternative concept avoids space dimension depending regularity assumptions for (quantum) physical variational model (wave package) states and solutions (defined e.g. by energy or operator minimization problems) and physical problem specific "force" types.

The common distributional Hilbert space framework is also proposed for a proof of the Landau damping alternatively to the approach from C. Villani. Our approach basically replaces an analysis of the classical (strong) partial differential (Vlasov) equation (PDE) in a corresponding Banach space framework by a quantum field theory adequate (weak) variational representation of the concerned PDE system. This goes along with a corresponding replacement of the “hybrid” and “gliding” analytical norms (taking into account the transfer of regularity to small velocity scales) by problem adequate Hilbert space norms H(-1/2) resp. H(1/2). The latter ones enable a "fermions quantum state" Hilbert space H(0), which is dense in H(-1/2) with respect to the H(-1/2) norm, and its related (orthogonal) "bosons quantum state" Hilbert space H(-1/2)-H(0), which is a closed subspace of H(-1/2).


Braun K., A distributional Hilbert space framework to prove the Landau damping phenomenon

                         (October 15, 2018, update : pp. 2,3,6)

As a shortcut reference to the underlying mathematical principles of classical fluid mechanics we refer to

Serrin J., Mathematical Principles of Classical Fluid Mechanics