Members of the drivers union Aslef are due to walk out for three days in the dispute over driver-only trains.
Southern's parent company Govia Thameslink (GTR) has asked the Court of Appeal to overturn last week's High Court refusal to ban the strike action.
A decision will be given at the end of the hearing, due to last all day.
Lord Justice Elias, sitting with Lord Justice Lewison and Lord Justice Lloyd Jones, have been asked to overturn the refusal to grant an injunction blocking the "unprecedented" action.
GTR has already been hit by a series of strikes by the RMT union in a long-running dispute over the role of conductors.
Drivers are due to strike on Tuesday, Wednesday and Friday, with a week-long walkout planned from 9 to 14 January.
Southern said given the timing of the Court of Appeal's decision, it would be too late to get trains and drivers in the correct position for Tuesday, and any services it could run would be "extremely limited".
Angie Doll, passenger services director, said: "We hope for the sake of our passengers our application to the Court of Appeal is successful.
"Even if we are able to stop the strikes through the court, services will still be very heavily  impacted tomorrow.
"We will work through the night to try and provide as many services as possible, but we are still advising passengers not to travel as we will not be able to offer a robust service they can rely on.
"We are sorry but the industrial action by Aslef leaves us no viable alternative, but be assured we will do all we can overnight," she said.
The RMT union, which has been staging strikes since April, is also planning further stoppages either side of Christmas.
00:01: Tuesday 13 December to 23:59 Wednesday 14 December (Aslef)
00:01 Friday 16 December to 23:59: Friday 16 December (Aslef)
00:01 Monday 19 December to 23:59 Tuesday 20 December (RMT)
00:01 Saturday 31 December to 23:59 Monday 2 January (RMT)
00:01 Monday 9 January to 23:59: Saturday 14 January (Aslef)

The duchess was offered a posy from 22-month-old Elizabeth Colquhoun, who then decided she wanted to keep the bouquet and refused to hand it over.
Camilla was visiting Perth's state library on the final day of the royal tour of Australia and New Zealand.
Elizabeth's mother Chantelle Colquhoun said her daughter was "strong willed".
Originally from the Cornish town of Camborne, Mrs Colquhoun, 39, was in a fit of giggles at her daughter's antics and said she was "just the same way at home".
The Duchess of Cornwall eventually managed to take the bouquet from Elizabeth and playfully held it just out of the grasp of the youngster until she grabbed it.
The duchess was at a book reading session at the time.
Meanwhile, the Prince of Wales was given a tour of a conservation centre.
He met students at the Western Australia Seed Technology Centre in Perth after attending a service at St George's Cathedral in the city with the duchess.
The royal couple have spent 12 days touring Australia and New Zealand.
Prince Charles celebrated his 67th birthday on Saturday with an Australian beach barbecue at Cottlesloe beach in Perth where he was joined by 500 guests.
Strike-hit rail operator Southern has warned passengers not to travel on Tuesday, irrespective of the outcome of a last-ditch bid to halt a walkout.
[NEXT_CONCEPT]
The Duchess of Cornwall found herself in a playful tug of war with a little girl determined to hang onto her bunch of flowers.