You were assigning $query when null, but you should've assigned $car_list which is the original query you're building on.
public function list(Request $request){
$carGov = $request->get('carGov');
$carName = $request->get('carName');
$carModel = $request->get('carModel');
$car_list = Car::select('cargov', 'carname', 'carmodel');
$car_list = is_null($carModel) ? $car_list : $car_list->wherecarmodel($carModel);
$car_list = is_null($carName) ? $car_list : $car_list->wherecarname($carName);
$result = $car_list->orderBy('updated_at','DESC')->paginate(10);
return response()->json($result,200);
}