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On a common compact solution domain, a right-hand side continuous in parameters and uniformly Lipschitz in the state yields continuous dependence of the solution on those parameters. The Gronwall inequality controls the difference of two solutions. A transverse zero of an event function persists under a small parameter change, which allows a simple stellar surface in the Newtonian limit to persist in relativistic models.

An energy estimate tests a differential equation against the solution or one of its derivatives to control a norm by initial data and forcing. Coercive terms represent dissipation, while Young inequality and the Gronwall inequality control lower-order terms.

For a zero-boundary centered semidiscretization $U^{\prime }=D_{h}U+V_{h}U$, where $D_h$ is the scaled Dirichlet discrete Laplacian and $V_h=\mathrm{diag}(a_j)$ with $a_j\le a_{+}$, summation by parts gives $(U,D_{h}U{)}_h\le 0$. Hence $\frac{1}{2}(d/dt)\Vert U{\Vert }_h^2\le a_{+}\Vert U{\Vert }_h^2$. The Gronwall inequality yields $\Vert U(t){\Vert }_h\le e^{a_{+}t}\Vert U(0){\Vert }_h$, a mesh-independent stability bound on every fixed finite interval. Positive reaction coefficients can permit physical growth without violating this form of stability.

Unit-mass solutions of the isotropic Maxwell molecule collision operator with matching first moments obey this comparison. The Bobylev identity, the Fourier bound by one, and $|{\xi }^{+}{|}^2+|{\xi }^{-}{|}^2=|\xi {|}^2$ give a scalar damped differential inequality. Its integral form and the Gronwall inequality yield nonexpansion in the Fourier distance of order two. This does not alone establish strict contraction or equilibrium convergence.

If $H$ is $C^2$ and its spatial Hessian matrix is uniformly bounded on every finite time interval, its Hamiltonian vector field is globally Lipschitz continuous in phase space with at most linear growth. The Picard-Lindelof theorem and Gronwall inequality give a unique Hamiltonian flow for all finite forward and backward times. Bounds on ${\nabla }_{z}H(t,0)$ follow from time continuity on compact intervals.

If $F\in C^1(\mathbb{R}\times {\mathbb{R}}^d)$ and $|F(t,x)|\le K(1+|x|)$, every characteristic curve is defined for all real time. The Gronwall inequality bounds $1+|X(t)|$ on each bounded time interval, so local ordinary differential equation existence cannot terminate through escape to infinity. Uniqueness and the variational equation give a global $C^1$ spatial flow diffeomorphism, with Jacobian $J>0$ satisfying $J_t=(\mathrm{div}F)(t,X)J$.

Along a smooth curve $\gamma :I\to M$, a horizontal lift of a linear connection on a vector bundle has fibre equation $v^{\prime }=-\Gamma (\dot{\gamma })v$. On each compact time interval the coefficient matrix is bounded, and the Gronwall inequality prevents finite-time blowup. A finite cover by vector bundle trivializations and ODE uniqueness patch local solutions. Exhausting the open interval gives a unique lift on all of $I$. Its endpoint map is linear, and reversal of the path gives its inverse; no completeness of the base manifold is needed.

For $u_{tt}-\Delta u+u^3=0$ in three spatial dimensions, conserved positive wave energy $E$ controls $\Vert \nabla u{\Vert }_2$. The Sobolev inequality gives $\Vert u^2\nabla u{\Vert }_2\lesssim \Vert \nabla u{\Vert }_2^2\Vert D^{2}u{\Vert }_2\lesssim E\Vert D^{2}u{\Vert }_2$. Differentiated wave energy estimates and the Gronwall inequality then give $\Vert u(t){\Vert }_{{\dot{H}}^2}+\Vert u_t(t){\Vert }_{{\dot{H}}^1}\le CD{e}^{CEt}$, where $D$ bounds the corresponding initial Sobolev norms.

If $2(x-y)(b(x)-b(y))+(\sigma (x)-\sigma (y){)}^2\le -k(x-y{)}^2$, the Itô formula for the squared difference and the Gronwall inequality give $\mathbb{E}|X_t-Y_t{|}^2\le e^{-kt}\mathbb{E}|X_0-Y_0{|}^2$. Localization of $e^{kt}|X_t-Y_t{|}^2$ as a nonnegative local supermartingale gives the same estimate under suitable existence hypotheses.

A one-sided Lipschitz condition controls the growth of distances between solutions of an ordinary differential equation. Differentiating the squared difference and applying the Gronwall inequality gives $\Vert y(t)-z(t)\Vert \le e^{\nu t}\Vert y(0)-z(0)\Vert$. The case $\nu \le 0$ is a dissipative vector field. Ordinary Lipschitz continuity implies such a bound, but the converse fails: $f(y)=-y^3$ has one-sided constant zero on the real line without being globally Lipschitz continuous.

If a continuous nondecreasing modulus $\mu$ satisfies this divergence condition, an estimate $|b(t,x)-b(t,y)|\le A(t)\mu (|x-y|)$ with locally integrable $A$ gives uniqueness of characteristic curves. Apply a regularized integral comparison to their separation. For the log-Lipschitz modulus, ${\delta }^{\prime }(t)\le A(t)\delta (1-\log \delta )$ gives $\delta (t)\le \exp (1-(1-\log \delta (0))\exp (-{\int }_0^{t}A))$ while $\delta \le 1$. Zero initial separation stays zero. Unlike the usual linear Gronwall inequality, this criterion allows non-Lipschitz vector fields.