Conditional LAD-to-LL residual and finite-interval stability

Use the flat definitions $f_0$, $B$, and ${\mathrm{\Pi }}_U$ of Reduced-order Landau–Lifshitz local ODE and mass shell. Let $I=[{\tau }_{-},{\tau }_{+}]$ with ${\tau }_{-}<{\tau }_{+}$ be a common compact proper-time interval, and suppose each curve below is $C^3$ on an open neighborhood of $I$. For each $0<\epsilon \le {\epsilon }_{\ast }$, let $x_{\epsilon }:I\to A$ be a curve in affine Minkowski space with $U_{\epsilon }={\dot{x}}_{\epsilon }$, $a_{\epsilon }={\dot{U}}_{\epsilon }$, and $\eta (U_{\epsilon },U_{\epsilon })=-c^2$. Let $F$ be a fixed $C^2$ two-form on a neighborhood of the curves, fix a constant ${\alpha }_{\epsilon }\in \mathbb{R}$ for each $\epsilon$, and set $f_{0,\epsilon }={\alpha }_{\epsilon }F(U_{\epsilon })$ and ${\delta }_{\epsilon }:=a_{\epsilon }-f_{0,\epsilon }$. Suppose every curve satisfies the flat LAD equation

$a_{\epsilon }=f_{0,\epsilon }+\epsilon ({\dot{a}}_{\epsilon }-\frac{\eta (a_{\epsilon },a_{\epsilon })}{c^2}{U}_{\epsilon })\mathrm{.}$

Use the Euclidean norm of inertial components and its induced operator norm. In the SI interpretation $\epsilon$ is a time and ${\alpha }_{\epsilon }=q\mathrm{/}m$; all constants and field norms carry the corresponding units. Assume constants $A,M,P,C_0,C_1\ge 0$, independent of $\epsilon$, satisfy on $I$

$\mathrm{\mid }{\alpha }_{\epsilon }\mathrm{\mid }\le A,\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le M,\mathrm{\parallel }{\mathrm{\Pi }}_{U_{\epsilon }}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le P,\mathrm{\parallel }{\delta }_{\epsilon }\mathrm{\parallel }\le C_0\epsilon ,\mathrm{\parallel }{\dot{\delta }}_{\epsilon }\mathrm{\parallel }\le C_1\epsilon \mathrm{.}$

Then throughout $I$,

$\parallel a_{\epsilon }-(f_{0,\epsilon }+\epsilon {\mathrm{\Pi }}_{U_{\epsilon }}{B}_{\epsilon })\parallel \le P(C_1+AM{C}_0){\epsilon }^2\mathrm{.}$

The estimate is conditional on the displayed common-interval and uniform bounds. It does not establish that regular LAD solutions satisfying them exist, nor does it imply global-in-time convergence.

For a separate finite-interval comparison, let $d\in \mathbb{N}$ with $d\ge 1$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^d\times {\mathbb{R}}^d$ be open in the topology of the usual Euclidean metric $d_2$ on ${\mathbb{R}}^{2d}$, and let $K\subseteq \mathrm{\Omega }$ be compact for that metric. Let $T>0$ and ${\tau }_e,C,L\ge 0$. Fix positive coordinate scales $s_j$ for the $2d$ phase coordinates and define $\mathrm{\parallel }\xi {\mathrm{\parallel }}_1:={\sum }_{j=1}^{2d}\mathrm{\mid }{\xi }_j\mathrm{\mid }\mathrm{/}{s}_j$ for $\xi \in {\mathbb{R}}^{2d}$. In a dimensional application choose the position and velocity scales consistently with one fixed time unit, so the two blocks of the lifted vector fields below have the same rate units. All constants use these fixed scales; in particular $L$ has units of inverse time, $LT$ is dimensionless, and $C{\tau }_e^2$ has the units of a phase-space rate. Let $A_{{\tau }_e},{\tilde{A}}_{{\tau }_e}:\mathrm{\Omega }\to {\mathbb{R}}^d$ be specified acceleration maps, and define the corresponding second-order phase vector fields

$X_{{\tau }_e}(q,v):=(v,A_{{\tau }_e}(q,v)),{\tilde{X}}_{{\tau }_e}(q,v):=(v,{\tilde{A}}_{{\tau }_e}(q,v))\mathrm{.}$

All vector integrals in this comparison are taken componentwise as real Riemann integrals.

Assume that for all $z,z^{\mathrm{\prime }}\in K$,

$\mathrm{\parallel }{X}_{{\tau }_e}(z)-X_{{\tau }_e}(z^{\mathrm{\prime }}){\mathrm{\parallel }}_1\le L\mathrm{\parallel }z-z^{\mathrm{\prime }}{\mathrm{\parallel }}_1,\mathrm{\parallel }{\tilde{X}}_{{\tau }_e}(z)-{\tilde{X}}_{{\tau }_e}(z^{\mathrm{\prime }}){\mathrm{\parallel }}_1\le L\mathrm{\parallel }z-z^{\mathrm{\prime }}{\mathrm{\parallel }}_1,$

and that $\mathrm{\parallel }{X}_{{\tau }_e}(z)-{\tilde{X}}_{{\tau }_e}(z){\mathrm{\parallel }}_1\le C{\tau }_e^2$ for every $z\in K$. If $z_1,z_2:[0,T]\to K$ are restrictions of $C^1$ maps on an open interval containing $[0,T]$, solve ${\dot{z}}_1=X_{{\tau }_e}(z_1)$ and ${\dot{z}}_2={\tilde{X}}_{{\tau }_e}(z_2)$ pointwise on $[0,T]$, and satisfy $z_1(0)=z_2(0)$, then

${\sup }_{0\le t\le T}\mathrm{\parallel }{z}_1(t)-z_2(t){\mathrm{\parallel }}_1\le C{\tau }_e^{2}T{e}^{LT}\mathrm{.}$

This stability claim assumes both solutions exist on the full interval and remain in the stated common region. It concerns only the two specified second-order systems; it is not an existence or convergence claim for the third-order LAD equation.