Is there latexmk functionality for LuaTeX?

You can use Lua(La)TeX with latexmk by setting the -pdflatex parameter:

-pdflatex=<program> - set program used for pdflatex.
(replace '<program>' by the program name)

So latexmk should call lualatex instead of pdflatex if you invoke it like this:

latexmk -pdflatex=lualatex -pdf <texfile>

or with newer versions of latexmk there is direct Lua(La)TeX/XeLaTeX support:

-lualatex
Use lualatex. That is, use lualatex to process the source file(s) to pdf (in place of pdflatex).
This option is exactly equivalent to specifying the following sequence of options:
-pdflatex="lualatex %O %S" -pdf -dvi- -ps-

Instead of adding the -lualatex flag to each application of latexmk, you could place the following

$pdflatex = 'lualatex -file-line-error %O %S';
$pdf_mode = 1;

in an '.latexmkrc' file. This file should reside in your home folder, if you use lualatex in all your jobs, or in the folder(s) where your lualatex jobs are.

More generally, you will find a host of examples and boilerlate code for .latexmkrc in the respective CTAN repository

Since version 4.51 of latexmk, you can use the following .latexmkrc:

$pdf_mode = 4;
$postscript_mode = $dvi_mode = 0;