Jeremy Renner, Facing 'Long Road to Recovery,' Is 'Overwhelmed By the Showing of Love and Support': Source

Jeremy Renner is on the mend following his horrific New Year's Day snowplow accident, but he faces "a long road to recovery," a source reveals in the new issue of PEOPLE.

"Jeremy is making positive progress," continues the source, echoing a sentiment from Renner's sister Kym, who also told PEOPLE that the 52-year-old star of The Avengers was "crushing all progress goals."

The actor has been hospitalized since Jan. 1, when he was airlifted to a local medical center. He suffered "blunt chest trauma and orthopedic injuries," according to a statement from his rep, and underwent emergency surgery the next day.

Since then, fans and famous friends alike—including Evangeline Lilly, Hailee Steinfeld and Chris Hemsworth have all wished the actor well in his recovery, with Lilly, 43, writing on Instagram that Renner has "always been one of the most grounded and real people I ever met in Hollywood."

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The messages have warmed Renner's heart as he embarks on the arduous journey of healing. Says the source, "He is overwhelmed by the showing of love and support."

Jeremy Renner/Instagram

Renner has expressed gratitude in several Instagram posts. Renner gave his first update to fans on Jan. 3, posting a hospital bed selfie to Instagram and thanking well-wishers for their support. "Thank you all for your kind words. ??. Im too messed up now to type. But I send love to you all."

Two days later, he shared a video to his Instagram Story where his sister could be seen massaging his head and making her brother laugh as their mother looked on during what the actor wrote was a "spa moment to lift my spirits."

On Jan. 6, he posted an image on his Instagram Story surrounded by hospital staff alongside a message of gratitude. "Thank you renowned medical ICU team for beginning this journey," he wrote alongside the image with a series of prayer emojis beneath the photo.

RELATED: Jeremy Renner Thanks 'Renowned' Medical Staff in Another Recovery Update from the Hospital

Jesse Grant/Getty Jeremy Renner

Renner, who celebrated his 52nd birthday in the hospital Jan. 7, was "moving snow from his driveway so that his family members could depart his home after ringing in the new year together," his rep previously told PEOPLE. "He was also helping clear out the snow of his neighbor's home as everyone up there had been without power for 24 hours, and there had been a large snowfall."

Meanwhile, Washoe County Sheriff Darin Balaam said in a press conference the same day that Renner was helping a family member get a "stuck" vehicle out of the estimated 3 feet of snowfall from the night before when the incident occurred.

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"Mr. Renner went to retrieve his PistenBully, or snowcat — an extremely large piece of snow-removal equipment weighing at least 14,330 pounds — in an effort to get his vehicle moving. After successfully towing his personal vehicle from its stuck location, Mr. Renner got out of his PistenBully to speak to his family member," said Balaam.

"At this point, it is observed that the PistenBully started to roll. In an effort to stop the rolling PistenBully, Mr. Renner attempts to get back into the driver's seat of the PistenBully. Based on our investigation, it's at this point that Mr. Renner is run over by the PistenBully."

For more on Jeremy Renner, pick up the new issue of PEOPLE.