The Coast Guard suspended its rescue mission around 9 p.m. Thursday.
Aguiar's 31-foot fishing boat, the T.T. Zion, washed onto Fort Lauderdale beach about 1:15 a.m. Wednesday, its lights on and twin outboard engines running.
Fort Lauderdale Police Detective Travis Mandell said Thursday investigators believe Aguiar got in the boat alone, "but that's not to say he didn't meet up with anyone."
Ellen Aguiar's petition says her son "disappeared as the result of mental derangement or other mental cause" or disappeared "under circumstances indicating that he may have died, either naturally, accidentally or at the hands of another." It further states that he suffers from "severe bipolar disorder."
The missing man's cellphone and wallet were found on the boat by police, the 59-year-old Ellen Aguiar, of Pompano Beach, says in her filing.
He was last seen by the family nanny at home around 7 p.m. Tuesday, the petition states.
"Everyone is trying to figure out why he got in that boat," Mandell said. The National Weather Service had issued a small-craft warning for winds of 15 to 20 knots, seas up to 5 feet and isolated thunderstorms.
The Brazilian-born Aguiar made his fortune in oil and gas. He attended Westminster Academy Christian school in Fort Lauderdale, and spent one year at Clemson University as a business management major on a tennis scholarship before he left in 1999.
He and his wife Jaime, 33, have three boys and one girl, ages 7 to 10 months.
Court documents also indicate that in July 2011, Aguiar's wife and mother successfully petitioned a Miami-Dade judge to appoint an emergency guardian for him for 90 days.
In 2009, Aguiar claimed Broward sheriff's deputies beat him after a traffic stop and made anti-Semitic remarks, comments an agency spokesman called "ludicrous."
Aguiar pleaded no contest to two misdemeanor drug charges arising from the traffic stop and avoided serving time on probation or in jail.
He has a $50 million investment portfolio: $15 million in cash and $35 million in Israeli real estate, according to his mother's emergency petition.
"There is an imminent danger that the property of the absentee is in danger of being wasted, misappropriated or lost unless immediate action is taken because the absentee has disappeared … and assets must be protected from waste and/or dissipation," the petition states.
Aguiar seeks support for her son's spouse and children, and the authority to respond to lawsuits, manage his properties and pay mortgages and employees' salaries.
Aguiar denied making this claim.