Well, "private enterprise" is not what I call "capitalism". In fact, it is not what any consequent Marxist would call "capitalism". To us, capitalism is a system where production is controlled by capital, and capital is self-agrandizing money.
"Private enterprise" can be of a capitalist nature, if and only if society as a whole is capitalist. Private enterprises such as that of an slaveowner in Classic Athens (or modern Louisiannia), or such as that of a medieval guild member, or the modern owner of the shop around the corner are not capitalist.
To us, capitalism either is or is not; there is no much sence in expressions like "more or less capitalist", "a bit of capitalism and a bit of socialism", etc. Either capital controls production, and it is capitalism, or it doesn't, and it is not capitalism.
By "private enterprise" I suppose you mean economic private enterprise, ie, the production of commodities for sale in a market system. This is likely to survive for a significant period of time after a socialist revolution. Our point is not to abolish the markets as an end of itself, but to replace markets with more efficient mechanisms of distribution. Merely suppressing markets by decree is likely to result in even less efficient distribution of goods until higher forms of socialisation are in place.
What isn't likely to survive a revolution for much time is the private property of corporations. These will likely be socialised, meaning workers' control of the workplace and social control of the destination of their economic output, through democratically elected committees.
Is that what you are asking about?
Luís Henrique