McKean Entropy-Production Conjecture

For the space-homogeneous Boltzmann equation in dimension $3$ with collision kernels $$B=\frac{1}{4\pi}|v-v_*|^\gamma,\qquad \gamma\in[0,1],$$ the paper constructs smooth, positive, radial mixtures $f_R=(1-p)M_1+pM_R$ for which $\partial_tD(f_R)>0$ for sufficiently large $R$. Thus entropy production need not decrease even for Maxwell molecules ($\gamma=0$) or hard spheres ($\gamma=1$), negatively resolving McKean's monotonicity question for these physically standard kernels.

For the space-homogeneous Boltzmann equation in dimension $3$ with collision kernels $$B=\frac{1}{4\pi}|v-v_*|^\gamma,\qquad \gamma\in[0,1],$$ the paper constructs smooth, positive, radial mixtures $f_R=(1-p)M_1+pM_R$ for which $\partial_tD(f_R)>0$ for sufficiently large $R$. Thus entropy production need not decrease even for Maxwell molecules ($\gamma=0$) or hard spheres ($\gamma=1$), negatively resolving McKean's monotonicity question for these physically standard kernels.