= Solution
Use the intended regular scaling ansatz: $g(u)\sim C u^y$, $g'(u)\sim Cy u^{y-1}$ and $g''(u)=Cy(y-1)u^{y-2}+o(u^{y-2})$, with $C>0$, $y>0$. Relative to the drag magnitude $g'$, the two inertial contributions obey
$$
\frac{g''}{g'}\sim\frac{y-1}{u},\qquad \frac{g'^2/g}{g'}=\frac{g'}g\sim\frac yu.
$$
Thus both inertial-to-drag ratios tend to zero at large $u$, whatever the positive growth exponent; for $y=1$ the leading acceleration term vanishes identically. The long-time <drag-limited hydrodynamic coarsening> balance is therefore
$$
c_dg'\sim\frac{c_s}{g^2},\qquad g^3(u)\sim3\frac{c_s}{c_d}u,
$$
so
$$
\boxed{g(u)\sim\left(3c_su/c_d\right)^{1/3},\qquad L(t)\sim\left(3\frac{c_s}{c_d}\frac\sigma{\bar\eta}t\right)^{1/3}.}
$$
The leading law is independent of <mass density>. Within the drag-capillary reduction, finite initial data give $L^3(t)-L^3(t_*)=3(c_s/c_d)(\sigma/\bar\eta)(t-t_*)$.
The <derivative> hypotheses are important: bare $g(u)\sim u^y$ does not justify differentiating an asymptotic equivalence. For instance, $g(u)=u+\tfrac12\int_1^u\sin(s^3)\,ds$ is smooth and increasing, with $g(u)/u\to1$, but $g''=\tfrac32u^2\cos(u^3)$ is not small compared with $g'=1+\tfrac12\sin(u^3)$. This is a counterexample to an inference from the bare asymptotic relation, not a solution of the coarsening equation. The physical claim uses the regular self-similar power law, for which the ratio calculation above applies.
This conclusion concerns the model's advective drag-capillary channel. Diffusive <Ostwald ripening> can also have a $1/3$ growth exponent; equality of exponents does not establish that diffusion is negligible. If it is retained, the prefactor and crossover function may also depend on <order-parameter mobility>, and one must compare transport mechanisms rather than infer the mechanism from the exponent alone.
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