Continued Ukrainian advances would reverse the situation of a few months ago, putting the Russians inside Bakhmut at risk of being surrounded and trapped, and would demonstrate that the deep, fortified lines the Russians have built across Ukraine can be breached. Success around Bakhmut would also provide a major morale boost for Ukraine and a serious blow to Russia, denying it the only military achievement that for months had seemed within its grasp.

The possible reversal of fortunes comes as Ukraine is preparing to mount a broader counteroffensive, aiming for a dramatic breakthrough in a war that has settled into a grueling slugfest, with much blood spilled but little ground gained. While the dynamics around Bakhmut are somewhat specific to that battle, Ukrainian commanders say they hope to build on the lessons learned there when they try to attack in other places along the 600-mile front line.

“When you retreat it is very difficult to stop,” said Col. Andriy Biletsky, the commander of Ukraine’s 3rd Assault Brigade, whose soldiers made the first breakthrough of the Russian lines last week. “When you want to advance, it is very difficult to start.”

He cautioned that he was waiting to see if “there is a chain of five, six, seven victories” before assessing the state of the fighting, but he was hopeful.