  The present article investigates magnetic amplification in the upstream
medium of SNR blast wave through both resonant and non-resonant regimes of the
streaming instability. It aims at a better understanding of the diffusive shock
acceleration (DSA) efficiency considering various relaxation processes of the
magnetic fluctuations in the downstream medium. Multi-wavelength radiative
signatures coming from the SNR shock wave are used in order to put to the test
the different downstream turbulence relaxation models. We confirm the result of
Parizot et al (2006) that the maximum CR energies should not go well beyond PeV
energies in young SNRs where X-ray filaments are observed. In order to match
observational data, we derive an upper limit on the magnetic field amplitude
insuring that stochastic particle reacceleration remain inefficient.
Considering then, various magnetic relaxation processes, we present two
necessary conditions to achieve efficient acceleration and X-ray filaments in
SNRs: 1/the turbulence must fulfil the inequality $2-\beta-\delta_{\rm d} \ge
0$ where $\beta$ is the turbulence spectral index while $\delta_d$ is the
relaxation length energy power-law index; 2/the typical relaxation length has
to be of the order the X-ray rim size. We identify that Alv\'enic/fast
magnetosonic mode damping does fulfil all conditions while non-linear
Kolmogorov damping does not. Confronting previous relaxation processes to
observational data, we deduct that among our SNR sample, the older ones (SN1006
& G347.3-0.5) fail to verify all conditions which means that their X-ray
filaments are likely controlled by radiative losses. The younger SNRs, Cas A,
Tycho and Kepler, do pass all tests and we infer that the downstream magnetic
field amplitude is lying in the range of 200-300 $\mu$ Gauss.

High Energy Astrophysical Phenomena;Astrophysics of Galaxies