Are there fractal solutions in continuum mechanics?

Continuous media belongs to the continuum and could have an elaborate structure. So I guess there is a fractal solution. But at present, I do not know a specific example, so I would like to ask my friends who know about this: is there a fractal solution in continuum mechanics?

The details are as follows, I’ve read some references about chaos and fractals like Strogatz’s "Nonlinear Dynamics and Chaos". But in that book, the main focus is on a finite-dimensional dynamic system. But continuum mechanics has infinite dimensions. And I am curious about how to use continuum mechanics to describe fractals. As what is said in "https://en.wikipedia.org/wiki/Fractal": analytically, fractals are usually nowhere differentiable. In contrast, the equation of continuum mechanics in its strong form is usually PDE like the Navier-Stokes equation and elasticity equation. And what I want to know is that what kind of equations in continuum mechanics can describe fractals(nowhere differentiable) and are equations like the Navier-Stokes equation have some closed-form fractal solutions like Julia set and Mandelbrot set in complex dynamics.