A nonlinear kinetic theory of cosmic ray ( CR ) acceleration in supernova remnants is employed to investigate the properties of the remnant SN 1987A .
It is shown that a large downstream magnetic field B _ { d } \approx 10 mG is required to fit the existing observational data .
Such a strong field together with the strong shock modification due to CR backreaction provides the steep and concave radioemission spectrum and considerable synchrotron cooling of high energy electrons which diminish their X-ray synchrotron flux below the observed Chandra flux which has to be considered as an upper limit for nonthermal X-ray emission .
The expected \gamma -ray energy flux at TeV-energies at the current epoch is 2 \times 10 ^ { -13 } erg/ ( cm ^ { 2 } s ) .