2 added 693 characters in body; deleted 7 characters in body; added 221 characters in body; added 2 characters in body

Here is a survey by DeLellis and Szekelyhidi about instances of the "h-Principle" and where convex integration is used to construct low regularity solutions to many equations of fluid mechanics:

You can use the method of convex integration (at least, basically the same kind of convexity argument -- Gromov himself prefers not to call this convex integration) to construct paradoxical wild solutions to PDE. For example, there are bounded solutions to incompressible Euler which are in the energy space and can have any prescribed energy density $\frac{1}{2} |v|^2(x,t)$ (in particular, they can be compactly supported in space and time). This is a bit shocking because sufficiently regular solutions to Euler conserve energy. The fact that weak solutions need not conserve energy is tied to ideas regarding the theory of turbulence, which is the main motivation for all these studies.

If you read Springer or Gromov you may not immediately recognize the similiarities between the analysis convex integration and the topology/geometry version (for example, sometimes Baire category arguments are used in analysis)analysis to simplify the technical arguments, sometimes at the expense of some regularity in the solution). But the arguments closely parallel Nash's proof that short maps can be approximated by $C^1$ isometric embeddings, which is where the story of convex integration begins. More recent developments regarding isometric embeddings can be found in the references to the survey linked above. One main challenge regarding both Euler and the isometric embedding problem is to find the degree of regularity at which there is a transition from flexibility to rigidity.

Preceding the developments in fluid mechanics, these convex integration was also used by Kirchheim, Muller and Sverak to exhibit elliptic systems coming from Euler Lagrange equations with solutions that are Lipschitz but nowhere $C^1$ -- this flexibility result contrasts the result of Evans that minimizers of the same kinds of functionals are smooth off a closed set of measure 0. There are also many related investigations in the calculus of variations tied to the stability of differential inclusions $\nabla u \in K$, especially regarding how they arise in the mathematical theory of elasticitymaterials. For example, James and Ball presented the idea that if $u : \Omega \subseteq {\mathbb R}^3 \to {\mathbb R}^3$ is the configuration of a crystal, its deformation gradient $\nabla u$ minimizes free energy $\int_\Omega W(\nabla u) dx$ by taking values pointwise in the set $K$ of critical points of $W$. Muller's book "Variational Models for Microstructures and Phase Transitions" has more on this topic (for example regarding how you can explain microstructures as patterns which are "trying to minimize" such a functional), but I think this is a bit more distant from the original question. The relevance is only that convex integration can be used to produce wild solutions to $\nabla u \in K$; but here $K$ might even be a finite set, and $u$ is only Lipschitz, so it's fairly different from the topological setting.

1

You asked about the $h$-principle, but I'll say something about convex integration instead.

Here is a survey by DeLellis and Szekelyhidi about instances of the "h-Principle" and convex integration used to construct low regularity solutions to many equations of fluid mechanics:

http://arxiv.org/abs/1111.2700

These analytic results, however, are different in flavor to what you usually call the $h$-principle in topology and geometry. In topology a nontrivial instance of the $h$-principle might say something like "you can invert the sphere $S^2 \subseteq {\mathbb R}^3$ through a regular family of immersions"; what makes it non-trivial is that there could have been a topological obstruction to doing so (for instance, you can't invert $S^1 \subseteq {\mathbb R}^2$ because the inclusion map $i$ and $-i$ have different degrees). In these analytic results, you're not exactly interested in homotopies.

You can use the method of convex integration (at least, basically the same kind of convexity argument -- Gromov himself prefers not to call this convex integration) to construct paradoxical solutions to PDE. For example, there are bounded solutions to incompressible Euler which are in the energy space and can have any prescribed energy density $\frac{1}{2} |v|^2(x,t)$ (in particular, they can be compactly supported in space and time). This is a bit shocking because sufficiently regular solutions to Euler conserve energy. The fact that weak solutions need not conserve energy is tied to ideas regarding the theory of turbulence, which is the main motivation for all these studies.

If you read Springer or Gromov you may not immediately recognize the similiarities between the analysis convex integration and the topology/geometry version (for example, sometimes Baire category arguments are used in analysis). But the arguments closely parallel Nash's proof that short maps can be approximated by $C^1$ isometric embeddings, which is where the story of convex integration begins. More recent developments regarding isometric embeddings can be found in the references to the survey linked above. One main challenge regarding both Euler and the isometric embedding problem is to find the degree of regularity at which there is a transition from flexibility to rigidity.

Preceding the developments in fluid mechanics, these convex integration was also used by Kirchheim, Muller and Sverak to exhibit elliptic systems coming from Euler Lagrange equations with solutions that are Lipschitz but nowhere $C^1$ -- this flexibility result contrasts the result of Evans that minimizers of the same kinds of functionals are smooth off a closed set of measure 0. There are also many related investigations in the calculus of variations tied to the stability of differential inclusions $\nabla u \in K$, especially regarding how they arise in the mathematical theory of elasticity.