= Solution
For an analytic, local <Landau theory> with independent global rotations of the <magnetization> vector, a uniform scalar potential can depend only on $\rho=\mathbf m\cdot\mathbf m$. Keeping terms through fourth order in the field, dropping an additive constant, and retaining the leading isotropic positive gradient term gives the displayed quadratic-plus-quartic functional. The stiffness has been absorbed into the <normalization> of length and field. Higher powers and higher derivatives are neglected in this long-wavelength approximation.
The quartic truncation is stable at large $|\mathbf m|$ when $u>0$. The coefficient $t$ changes sign at the mean-field critical point; close to it one commonly writes $t\propto T-T_c$. If $u<0$, higher stabilizing powers are essential, so the displayed truncation alone cannot determine a stable first-order phase diagram. For $u=0$ and $t<0$ it is likewise unbounded.
For $u>0$, spatially uniform stationary points obey $(t+4u\rho)\mathbf m=0$. The global minima are
$$
\boxed{\mathbf m=0\quad(t\geq0),\qquad
|\mathbf m|^2=-\frac{t}{4u}\quad(t<0).}
$$
The corresponding minimum free-energy density is zero and $-t^2/(16u)$, respectively. The disordered phase gives way continuously to a ferromagnet with an arbitrary direction of <magnetization>. At this mean-field transition the amplitude vanishes as $(-t)^{1/2}$; long-wavelength critical fluctuations can modify mean-field exponents in three dimensions.<Statistical Field Theory by David Tong>[§4.2, vector order parameter and rotationally invariant Landau expansion]{ref}
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