In the current standoff, Senate Republicans, who are in the minority, have asked House Republicans, in the majority, to refuse to adjourn. But the endgame here actually favors the White House, Jonathan Berenstein points to a Public Citizen article on HuffPo that grabs the relevant bit of the Constitution.
[The President] shall from time to time give to the Congress Information of the State of the Union, and recommend to their Consideration such Measures as he shall judge necessary and expedient; he may, on extraordinary Occasions, convene both Houses, or either of them, and in Case of Disagreement between them, with Respect to the Time of Adjournment, he may adjourn them to such Time as he shall think proper; he shall receive Ambassadors and other public Ministers; he shall take Care that the Laws be faithfully executed, and shall Commission all the Officers of the United States.(emphasis mine) When one considers that one of the primary grievances against King George was that he had dissolved colonial legislatures, the fact that the President has any power to adjourn the legislative branch for an unspecified length of time is a bit astounding. But in our present circumstances it might come in handy as a way to end disputes over appointments.
... if anyone can work up the gumption to use it. Instead it appears we may just declare the debt ceiling unconstitutional and move on.