Several of Ukraine’s claimed advances are concentrated along the Mokri Yaly, southwest of the town of Velyka Novosilka, in the Donetsk region. Ukraine’s military and a volunteer force fighting alongside it have reported capturing several small places there on both sides of the river, including, on Monday, the settlement of Storozhove.

In addition, there has been heavy fighting, and some claims of incremental Ukrainian advances, along an arc of more than 150 miles, from Bakhmut, to the northeast, to near the city of Kamianske, on the Dnipro, in the Zaporizhzhia region to the west.

A video posted online Monday and verified by The Times shows Ukrainian troops posing with a Ukrainian flag and walking around Storozhove. In another video verified by The Times, posted Sunday, Ukrainian soldiers place a flag inside a damaged building in Blahodatne, a village just across the river.

These videos and others are posted to the social media accounts of the Ukrainian units involved in the operation in an apparent effort to announce the retaking of each village. While the videos do show Ukrainian forces apparently operating freely in each location, it is not clear exactly how much control they have established. In several of the videos, apparent explosions and small-arms fire can be heard in the background.

Despite the modesty of the claims, they have buoyed the mood of Ukrainians. Among soldiers interviewed on Sunday at cafes in the city of Zaporizhzhia after they came off the front, there was a sense of momentum in the early advances. But they also described intense artillery barrages from the Russians.