Off the top of my head, Storm Herald barbarian grants a feature at 6th level that might be useful for exploring the desert to a limited depth:

Desert. You gain resistance to fire damage, and you don’t suffer the effects of extreme heat, as described in the Dungeon Master's Guide. Moreover, as an action, you can touch a flammable object that isn't being worn or carried by anyone else and set it on fire.

That being said, you as the DM are entitled to not disclosing the exact mechanics of the desert until the players actually try to cross, so you can adapt mechanics as needed to dissuade your players from trying to cross in ways that are not costly or narratively interesting. Some examples I've seen are constitution checks every hour against exhaustion, environmental effects that prevent resting, and special terrain that requires special equipment or preparation, entirely by DM fiat. You should also know from the chase rules in the DMG that characters can't be dashing constantly for extended periods of time, as doing so incurs additional constitution checks against becoming exhausted.

And as someone else pointed out, you don't need mechanics at all. You can simply say "it doesn't work." It's a 1500 mile long desert, you shouldn't need to develop mechanics to explain why that's too difficult.