15. Find the smallest constant $\lambda$, such that the inequality $x y\left(x^{2}+y^{2}\right)+y z\left(y^{2}+z^{2}\right)+z x\left(z^{2}+x^{2}\right) \leqslant \lambda(x+y+z)^{4}$ holds for all non-negative real numbers $x, y, z$.
$$
\begin{array}{l}
(x+y+z)^{4}=\left(x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x\right)^{2} \\
\geqslant 4\left(x^{2}+y^{2}+z^{2}\right)(2 x y+2 y z+2 z x)=8\left(x^{2}+y^{2}+z^{2}\right)(x y+y z+z x) \\
=8\left[x y\left(x^{2}+y^{2}\right)+y z\left(y^{2}+z^{2}\right)+z x\left(z^{2}+x^{2}\right)+x y z(x+y+z)\right]
\end{array}
$$
$\geqslant 8\left[x y\left(x^{2}+y^{2}\right)+y z\left(y^{2}+z^{2}\right)+z x\left(z^{2}+x^{2}\right)\right]$, equality holds when 2 of $x, y, z$ are equal, and 1 is 0. Therefore, the minimum value of $\lambda$ is $\frac{1}{8}$.