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 ����and I=3/2 K ��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 ��K^* \to ��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 ����\to K^* K^* and remarkably large for the reaction of I=1 ����\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.

26 pages, 15 figures