Now they did not know when they were going to get home. Ms. Vergara said she planned to spend the night on the airport floor, in hopes of buying a ticket to the mainland first thing in the morning.
“Whatever it takes,” she said.
The troubles for the airport, where there were more than 4.3 million outbound passengers last year, began after Hurricane Maria “destroyed or disabled” radars and navigational aids, according to the Federal Aviation Administration.
The F.A.A. said its recovery efforts could only move so quickly: Replacements had to be brought to Puerto Rico by land or by sea, and workers struggled this week to get to a long-range radar site nestled deep inside a national park. Technicians, the F.A.A. said, had found the final miles to the site “impassable” and had to use chain saws to bring in personnel and equipment.
The radar system was operational now, officials said, and additional air traffic frequencies were functioning. Gov. Ricardo Rosselló said that air traffic control in Puerto Rico was working at about a 20 percent clip, and two other airports, in Ceiba and Aguadilla, had been opened.
Although the airport in San Juan was accommodating 36 flights per hour — 18 arrivals and 18 departures — on Tuesday, commercial transport was limited because of the many military and relief flights. Federal officials said the total number was a sharp increase from Monday, when about eight flights were operating per hour. Despite the improvements, there were many cancellations.
“We’re very committed to growing the operation as quickly as we can and being back to where we were,” said Jim Butler, an American Airlines executive who is helping to oversee the carrier’s response.
American, Delta, JetBlue and Spirit Airlines were among the carriers operating limited flights, often with larger-than-normal planes. American, Mr. Butler said, was using inbound flights to ferry supplies, and he said that some planes were hauling 50,000 pounds of cargo.
“Every day it gets a little better,” he said. “I was down there on Friday, and we had no power, no connectivity whatsoever. We were operating by satellite phone.”
There were clear signs, though, of strain on the airport. Southwest Airlines, which said its flights “remain suspended,” warned passengers on its website against coming to the airport.
Officials restricted flights to San Juan between 6 p.m. and 7 a.m. to military and relief operations. Airports on St. Croix, St. Martin and St. Thomas were closed entirely to airline traffic.
Even as many were struggling to get out, there were others clamoring to get in.
On Tuesday, an inbound Spirit flight was packed with more than 200 people, many of them in T-shirts and caps expressing Puerto Rican pride. Gilberto Gonzalez, 31, a chef who lives in Orlando, looked out the window as the plane flew over the island. Some of the damage was not evident; the skyscrapers of San Juan looked the same. But he also pointed to a cluster of modest houses that had been dashed by the storm.
Mr. Gonzalez had come to rebuild the roof that the storm had ripped off his grandparents’ house in the countryside east of San Juan. He said he was excited to be back.
“The island is going to come back better,” he said. “We’re going to rebuild it, stronger.”
The plane landed and the passengers erupted in a victorious roar. “Puerto Rico se levanta!” a man yelled. Puerto Rico is rising.
At the gate, the arriving passengers filed past another group of 200 or so. They either had a ticket out, to Fort Lauderdale, on the last flight of the night, or were standing by for the next available plane.
There were many long faces, slicked with sweat.
“We’re running for our life,” said Michelle Juarbe, 29, who was with her 6-year-old son and her boyfriend, José Carrion. “There’s help coming, but it’s not being felt in the streets yet. For us, it’s not safe.”
They had come from Isabela, a municipality on the northwestern coast. It is far from the seat of government in San Juan, and Ms. Juarbe said that the looting was bad enough to make her fear for her safety, and her son’s.
“They’re robbing houses, and they’re robbing where people live,” she said. “Once we got into the gate, I was finally able to calm down, because I knew we were leaving.”
A few feet away, Dianny Betancourt was sitting with his wife, Mirelys Ocasio, who was fanning their 2-year-old daughter, Laila. The couple, both 33, were flying standby. If they could not get out Tuesday they said they would try again, and again. They were unsure whether their flight out was the start of a temporary or a permanent relocation. Their house had been destroyed by wind and falling trees.
“I love my island,” Ms. Ocasio said, “but we need a future.”
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