# nLab differential equation

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Variational calculus

variational calculus

# Contents

## Idea

A differential equation is an equation involving terms that are derivatives (or differentials). One sometimes distinguishes partial differential equations (which involve partial derivatives) from ordinary differential equations (which don't).

## As sub-$\infty$-groupoids of tangent and jet bundle

Analogous to how ordinary equations determine and are determined by their spaces of solutions – the corresponding schemes – accordingly differential equations $F(\partial^i x)$ on sections of a bundle $E \to X$ determine and are determined by their solution spaces, which are sub-D-schemes of the jet bundle D-scheme:

$\array{ Solutions(F) &&\hookrightarrow&& Jet(E) \\ & \searrow && \swarrow \\ && \Im(X) }$

In fact there is an equivalence of categories between the Eilenberg-Moore category of the jet comonad over $X$ and the category of partial differential equations with variables in $X$ (Marvan 86).

At least in nice cases for differential equations on functions on $X$ this is equivalently modeled by sub-Lie algebroids of tangent Lie algebroids.

$\array{ Solutions(F) &&\hookrightarrow&& T X }$

see exterior differential system for details

## In terms of synthetic differential geometry

A perspective on differential equations from the nPOV of synthetic differential geometry is given in

• Outline of synthetic differential geometry , seminar notes (1998) (pdf)

### First order homogeneous ODEs as extension problems in a smooth topos

In a smooth topos with $X$ and $A$ any two objects and $D = \{x \in R | x^2 = 0\}$ the abstract tangent vector, a diagram, of the form

$\array{ {*}&\to& D &\stackrel{v}{\to}& X \\ &{}_{\mathllap{v(f)(x)}}\searrow& {}^{\mathllap{v(f)}}\downarrow & \swarrow_{\mathrlap{f}} \\ && A }$

• $f$ is a smooth function on $X$ with values in $A$;

• whose derivative along the tangent vector $v \in T_x X \subset T X = X^D$

• …is the tangent vector $v(f) \in T_{f(x)} A \subset T A = A^D$.

Accordingly, the diagram

$\array{ D &\stackrel{v}{\to}& X \\ {}_{\mathllap{\alpha}}\downarrow \\ A }$

may be read as encoding the differential equation $v(f)_x = \alpha$ (at one point) whose solutions $f \in A^X$ are the extensions that complete this diagram.

To get differential equations in the more common sense that they impose a condition on the derivative of $f$ at each single point of $X$ and varying smoothly with $X$, we think of $f : X \to A$ in terms of the exponential map

$f^X : X^X \to A^X$

and consider

$\array{ &&{*} \\ && \downarrow & \searrow^{\mathrlap{Id_X}} \\ {*}&\to& D &\stackrel{v}{\to}& X^X \\ &{}_{\mathllap{f}}\searrow&{}^{\mathllap{\alpha}}\downarrow & \swarrow_{\mathrlap{f^X}} \\ &&A^X } \,.$

In this diagram now

• $v : D \to X^X$ is the adjunct of a vector field $X \to T X = X^D$ on $X$;

(the commutativity of the top right triangle ensures that indeed this $X \to T X$ is a section of the tangent bundle projection $T X \to X$);

• $\alpha : D \to A^X$ is the adjunct of a map $X \to T A = A^D$ that sends each point $x \in X$ to a tangent vector in $T_{f(x)} A$

(enforced by the commutativity of the left bottom triangle)

• and the commutativity of the right lower triangle is the differential equation

$v(f) = \alpha \,.$

## References

General accounts include

• Sergiu Klainerman, PDE as a unified subject 2000 (pdf)

• Boris Kruglikov, Valentin Lychagin, Geometry of differential equations, pdf

• Arthemy Kiselev, The twelve lectures in the (non)commutative geometry of differential equations, preprint IHES M/12/13 pdf

• Yves Andre, Solution algebras of differential equations and quasi-homogeneous varieties, arXiv:1107.1179

• Mikhail Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986. x+363 pp.

• Peter Olver, Applications of Lie groups to differential equations, Springer; Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995.

Discussion of the spaces of solutions is in

• Batu Güneysu, Markus Pflaum, The profinite dimensional manifold structure of formal solution spaces of formally integrable PDE’s (arXiv:1308.1005)

Characterization of the category of partial differential equations as the Eilenberg-Moore category of coalgebras over the jet comonad is due to

• Michal Marvan, A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno (pdf)

In the language of D-modules and hence for the special case of linear differential equations, this appears as prop. 3.4.1.1 in

A domain specific programming language for differential equations is presented in

• Martin S. Alnaes, Anders Logg, Kristian B. Oelgaard, Marie E. Rognes, Garth N. Wells, Unified Form Language: A domain-specific language for weak formulations of partial differential equations, arXiv:1211.4047