Other than demonstrating the will to strike back, France's expanded air missions over Syria are pointless militarily and it is folly to think that France can push America to do something decisive in Syria:
France needed to demonstrate that it was willing and able to strike, and that the perpetrators should not feel too safe in their hideouts in the Middle East.
However, the real value of these strikes is negligible. A much larger bombing campaign against the Islamic State has been going on for months, with rather limited impact. Additional French firepower in this campaign is useful and welcome, but it won't turn a halfhearted campaign into a military success. Most other European countries are absent from the operation.
But even if more countries joined in, the net effect would be limited, as airpower alone is incapable of delivering the outcome that would make a difference: diminishing the foothold that the Islamic State has established in the Syrian and Iraqi lands it has occupied. That could be achieved only through a massive military buildup, including a significant number of ground troops. No one in Europe wants this. And even if the Europeans did want it, there is very little they could deploy. The options would be limited even if the political will and public support existed, which is not the case.
A serious military operation could really only be conducted in alliance with the United States, a country staunchly opposed to any armed entanglement on the ground in the region.
The French air strikes should merely be a placeholder until France can organize a bigger effort that includes ground power. And that effort should be aimed at defeating jihadis in Libya and eliminating the ISIL mini-state in central Libya along the coast in Khadaffi's former stronghold.
Cleaning up Libya and helping sane forces exercise control over the territory would have a lot of advantages.
It would deprive ISIL of a sanctuary.
It would slow down recruiting of jihadis.
It would gain control of the former Khadaffi arsenal.
It would end a source of instability to Egypt, Tunisia, Algeria, and points to the southwest in France's former colonies where France maintains security relations and troops to defend these countries.
It would slow down the flow of migrants heading to Europe from the south.
It would add oil production to the world supply.
French marines, paratroopers, and Foreign Legion troops would suffice to operate in the south to help suppress jihadi influence that spills out to neighbors (as it did in Mali after the fall of Khadaffi's government).
French forces could also operate out of the east or west where more reasonable Libyan elements dominate in order to clean out jihadis and drive into the central coastal areas to ultimately deny ISIL control of that region.
Ideally, France leverages relations with Saudi Arabia and Egypt (the Saudis bought for Egypt the Mistral amphibious vessels originally built for Russia) to get the Saudis to finance an Egyptian ground force of a mechanized division and special forces supported by air power to support Libyan forces in the east to defeat jihadis there.
France can support operations from the sea (move the French carrier there from the Gulf) and from NATO land bases in Sicily and Crete.
America could provide support as we did in the original Libya war.
Face it, France can't do the job against ISIL in their homeland spanning Iraq and Syria; and their contribution to any fight there is pointless. Only our level of effort can be decisive at this point. If we make the effort, France is irrelevant to success. And if we don't make the effort, France can't make the effort decisive with what it brings to the table.
Short of nuking Raqqa, of course.
But France has good troops who can make a difference on a front where smaller numbers can be decisive. France showed what they can do in Libya in 2011 and in Mali in 2013. What they need is a sector in the ISIL front where they can be decisive.
With forces in Libya that can be supported, and with Egyptian help that at best includes ground power, special forces, and air power (and at least includes special forces and air power), France can make a decisive effort against ISIL that will hurt the terrorist group and help stabilize a region of importance to France.
Open the French sector.
UPDATE: Speaking of Mali:
Islamist gunmen stormed a luxury hotel packed with foreigners in Mali's capital Bamako on Friday, taking 170 hostages in a former French colony that has been battling rebels allied with al Qaeda for several years.
A senior security source said some of the hostages had been freed after being made to recite verses from the Koran. The French newspaper Le Monde quoted the Malian security ministry as saying at least three hostages had been killed.
This is still going on. I dare say settling Libya down and securing its territory would help reduce the scope of the threat in Mali and the region.
UPDATE: Strategypage discusses Libya:
The mess in Libya is not helped by the presence of ISIL (al Qaeda in Iraq and the Levant). ISIL has found Libya an excellent place to be. For one thing ISIL can finance its operations in Libya largely through people smuggling. This is a growth industry, going from practically nothing in 2011 to over thousand paying illegal migrants a day. ...
ISIL attracts the more fanatic men from other militias and has concentrated this evil in a few places (Sirte, Derna and Benghazi) and wherever ISIL is it is under attack by local militias.
And the Mali connection:
Across the southern border in Mali Libya based Islamic terrorists attacked a major hotel in the Mali capital and killed 19 people The Al Qaeda second-in-command in Libya, Mokhtar Belmokhtar, took credit for the attack. Belmokhtar also took credit for a similar attack on another Mali hotel in August. Belmokhtar was also behind the January 2013 natural gas facility attack in southern Algeria that got 37 workers killed. He has also survived several attempts to kill him and has a reputation for being elusive. He survived such attacks in 2013, 2014 and early 2015. Since 2013 his al Mourabitoun faction has been using bases in southern Libya and has been seen operating in northern Mali and Niger. The U.S. is offering a $5 million reward for information that would lead to the death or capture of Belmokhtar. Al Mourabitoun continues to survive in Libya because of the chaos there.
Seriously, France should focus on the Libya Front to battle ISIL rather than try to ramp up an effort in Syria that will be futile unless we decide to go all out.