  Meson-meson nonresonant reactions governed by the quark-interchange mechanism
are studied in a potential that is derived from QCD. S-wave elastic phase
shifts for I=2 \pi\pi and I=3/2 K \pi scattering are obtained with wave
functions determined by the central spin-independent term of the potential. The
reactions include inelastic scatterings of two mesons in the ground-state
pseudoscalar octet and the ground-state vector nonet. Cross sections for
reactions involving pion, rho, K and K^* indicate that mesonic interactions in
matter consisting of only K and K^* can be stronger than mesonic interactions
in matter consisting of only pions and rhos and the reaction of I=3/2 \pi K^*
\to \rho K is most important among the endothermic nonresonant reactions. By
the quark-interchange mechanism we can offer \sqrt s-dependences of phi
absorption cross sections in collisions with pion and rho and relevant average
cross sections what are very small for the reaction of I=1 \pi \phi \to K^* K^*
and remarkably large for the reaction of I=1 \rho \phi \to K^* K^*. It is found
from the \sqrt s-dependences of cross sections that rho and K^* creation cross
sections can be larger than their absorption cross sections, respectively.
