A system (1′) formed of a wheel comprising a rim (3′) equipped with an inflation valve (4′), a tire (2′) mounted on said rim of said wheel, and an airtight cellular foam insert (5′) placed inside the volume defined by the internal wall of the tire (2′) and the rim of the wheel (3′). The foam insert (5′) is disposed in a first compartment (A), in communication with the atmosphere via a duct (44′, 46′), and isolated from the rest of the space inside the tire casing by a device (302′) ensuring the airtight seal between the wall of said first compartment (A) and an element of the internal wall of the rim or the casing, in such a way as to form a second airtight compartment (B) communicating with the inflation valve (4′) and inflated to tire utilisation pressure. The duct for providing communication with the atmosphere (44′, 46′) is equipped with a closing device (45′) which adopts the closed position during travel and is capable of being brought into the open position between two travel phases.