a hyperbolic initial value problem
... with a radiation operator definition at t=0
... and the Ritz-Galerkin method for the wave equation
The Maxwell equations build a symmetric hyperbolic system. The Yang-Mills equations are a nonlinear generalization of the Maxwell equations which is semi-linear. The Euler and Einstein equations are quasi-linear, but not semi-linear hyperbolic equations. Even the simplest linear hyperbolic equation (the wave equation) shows different ('not optimal') properties of shift theorems in standard Sobolev spaces.
There is a conceptual difference between elliptic and parabolic problem versus hyperbolic problems.
With respect to hyperbolic problems the adequate energy Hilbert space cannot be more regular than the proposed H(1/2) (energy) Hilbert space for the NSE solution. We provide a counter example that it is most likely that any H(a) for any a>0 is not sufficient. An alternative (“exponential decay”) inner product is proposed with ‘optimal’ shift theorem behavior of the d’Alembert operator. The prize to be paid is an extension of current Hilbert scale inner products with polynomial decay coefficients to an inner product with exponential decay.
We present an 'optimal order' shift theorem for the wave equation with respect to this alternative ("hyperbolic") energy norm. At the same time we emphasis that there is only one quantum energy “measure” (as proposed in http://www.fuchs-braun.com/ ) defined by same quantum energy norm.
We note the elegant role of the proposed NSE-H(1/2) energy norm in universal Teichmüller theory.
We further mention that in current "quantum radiation theories" the Huygens principles is neglected.
We provide a characterization of the 4-dimensional Minkowski space which is about the fact that the differential of the Fourier transform of the uniform distribution of unit mass over the unit sphere at the origin vanishes.
Following the same concept as for the parabolic (heat equation) case (Nitsche/Wheeler) we sketch a proof of the L(infinite) boundedness of the Ritz operator in case of finite element approximation spaces for the solution of the wave equation. The analysis restricts to the space dimension m=1 and m=3. The required assumptions to the finite element approximation spaces are linear resp. quadratic finite elements in order to enable the Nitsche duality technique.
Here we are
"the objective world has only been constructed at the price of taking the self, that is, mind, out of it, remaking it; obviously, therefore, it can neither act on it nor be acted on by any of ist parts. ... If this problem of the action of mind on matter cannot be solved within the framework of our scientific representatiion of the objective world, where and how can it be solved?" - E. Schrödinger, Mind and Matter