Good day, I would be happy to assist you with this question.
There is no legislated obligation for employees to give notice upon resignation. There are cases however where the courts have looked at the "reasonableness" of notice or the lack thereof. This usually comes into play in situations like yours, where employees have left with little to no notice and take up employment with a competitor or start their own competing business.
If an employee has failed to give reasonable notice considering all circumstances, then the employer can seek damages through the courts, as you are inquiring about. The first thing to figure out is what if any damages you have incurred as a result of the lack of notice or any breach of a fiduciary duty that the employee owes to the employer, including any breach of a non-competition clause in the working agreement.
Did the employee fail to complete work that he was assigned and you lost customers or contracts? Has the employee taken any clients from you? Was your business unable to operate due to him leaving, thereby causing you to suffer a loss?
You are certainly within your rights to sue him for damages, if you are able to identify specific damages that are a direct result of his failure to give reasonable notice. Also, as I indicated, you can sue if he is breaching a non-competition clause.
I look forward to hearing back from you so that we can discuss this matter further and determine what damages you have incurred and whether it's worth your while to pursue an action against him.