First here’s a situation map from Wikipedia, 2022 Russian invasion of Ukraine, as of 20 Mar 2022:
Now is a good time to look at the current situation in the Russian invasion of Ukraine, and see what might happen.
Three weeks into the war, Russia is still advancing but slowly. Both sides have lost maybe 5000 military dead and maybe the same number of wounded. (There have also been many civilian casualties, especially due to Russia indiscriminately shelling cities as a terror tactic to force Ukraine to capitulate. This is obviously evil, but not the topic of this post so I will not speak of it further here except insofar as it affects the military situation).
So, the military situation right now is either a stalemate or a slow Russian advance.
Reinforcements
The West is helping Ukraine by supplying equipment. Foreign volunteers from all over the world are also joining Ukraine. Russia can also move more troops and equipment into Ukraine. but the troops/equipment they've invaded with is probably their best so that limits their ability to reinforce. Russia is also talking about arming Syrian volunteers to fight Ukraine.
Regarding foreign volunteers, they're a mixed bag. Untrained volunteers will be of little use until they are properly trained. The most useful volunteers will be ones who've been trained by a first-rate military: this will be true of a significant number of Ukraine's volunteers but a much lower proportion of Russia's volunteers, since Syria doesn't a first-rate army (by first-rate army I have in mind what Bret Devereaux calls a "modern system" army).
Regarding equipment, the West manufactures a wide range of sophisticated military equipment some of which (Stinger anti-aircraft missiles and Javelin anti-tank missiles) has already been sent to Ukraine. But some of the more complex systems, such as long-range anti-aircraft missiles, might be unfamiliar to Ukrainian users and would take a long time to train them. So if long-range SAMs are sent to Ukraine they may well be the sort of former USSR systems that Ukrainian soliders are already familiar with.
Russia has large stocks of equipment at its disposal, but they are already using most of their best equipment in Ukraine so that limits what they can send. The West has vastly more money than Russia, and also a broader network of allies than Russia does, and therefore has significantly more ability to buy military equipment from all over the world (including former USSR equipment which Ukraine's military is familiar with, which was widely exported and is used by many countries).
So in terms of equipment supplies, the Ukrainian army has a lot more potential to receive further supplies than Russian forces do, presuming the West is prepared to keep sending them. And this is where Russian war crimes come in: the more war crimes Russia commits, and the more they are publicised (as they will be) the more pressure there will be on Western governments, from their own peoples, to help Ukraine. Putin calculates that murdering civilians will help him win his war; but it is more likely to do the opposite.
Another problem for Russia is logistics. They are having trouble supplying their existing forces in Ukraine (about 180,000 soldiers formed into about 125 battalion tactical groups (BTGs)). If they increase the size of their army, their logistical problems will only get worse. Whereas if Ukraine starts running out of trucks to supply their troops, the West will presumably supply some more.
One unknown factor is China. For now, China has been neutral, but friendly to Russia. If China chooses to help Russia, they could supply much military equipment to them. The downside for China is that doing so would harm their relations with the West; they don't want to be put under the same sanctions that Russia is under, as they will harm the Chinese economy and make the West less dependent on Chinese goods (which weakens china's leverage in any future diplomatic situation).
So now, let's look at military outcomes (and estimate a likelihood for each one), projecting ahead the next 6 months.
Russian victory (20%)
Here Russia breaks through the Ukrainian defences and occupies all or most of the country. I see this as unlikely unless the West loses resolve and stops sending significant arms to Ukraine.
Continued stalemate (60%)
Here neither side can make significant advances and the front becomes mostly static. Defence is easier than attack, and in modern warfare large attacking formations can be spotted by drones and targetted by artillery which increases the advantage for the defence.
In this scenario, neither side is able to make large scale attacks but small scale ones may still happen.
Ukrainian victory (20%)
In this scenario Ukraine is able to push Russian forces out of all or most of the land they have so far conquered in this invasion (i.e. excluding Crimea and the Donetsk and Luhansk separatist republics). This might happen because of steadily increasing Ukrainian strength, and the Russian army hurt by attrition, soldiers surrendering, and low morale. One can imagine local Ukrainian offensives leading to a general Russian retreat and rout as their troops flee.
Peace talks
There are ongoing peace negotiations. Ukraine is reported to have conceded NATO membership in these negotiations which make Russia acceptance more likely.
For there to be a negotiated peace, there has to be a deal that both sides will agree to. So what are the minimum deals each side might agree to?
For Russia, Putin needs things that he can plausibly see as a victory. These might include:
(A) international acceptance of the separatist republics and Russia holding Crimea
(B) Ukraine agrees not to join NATO
(C) Ukraine agrees not to join EU
(D) NATO and EU countries agree to end sanctions against Russia
(E) Ukraine agrees to cede more territory to Russia
Ukraine's number one priority is security. What they want is a guarantee that Russia won't attack them again. Obviously they can't just take Putin's word that he won't as his word is worthless. But what would work for them is being heavily defended so that Putin realises future attacks would be futile. Ukraine would also want some guarantee that Russia won't make continual small pinprick attacks, hoping to wear Ukraine down. Ukraine also wants economic development, which is always a good long-term foundation for security, as money makes all things easier.
So Ukraine's demands might include:
(F) Russian withdrawal to the 2021 borders
(G) NATO/EU countries supply Ukraine with large quantities of weapons and economic assistance
(H) Ukraine allowed to join EU, or at least have access to EU markets through EEA
(I) international peacekeepers on the border with Russia to prevent future clashes
(J) NATO countries (especially USA) give security guarantee to Ukraine
Of these, several (notably A, D, G, H, I, and J) involve countries other than Russia and Ukraine, so I suspect that peace talks will only succeed if other countries are involved.
Do I think there will be a negotiated settlement within 6 months? The main difficulty is the sides don't trust each other. So I'd give it a 40% chance.
This marks the start of the 2nd Cold War
Regardless of who wins militarily or whether there is a negotiated settlement, this war changes everything and brings recent history into a new era.
The previous era, 1991-2022, was the America-led unipolar world, where the USA was the dominant power. Putin goin into power in 2000, and while people in the West (peoples and governments) might have thought thought Putin was bad, they also thought he was someone they can tolerate.
In the current era, 2022-whenever, is the 2nd Cold War between the West (led by the USA) and the Beijing-Moscow Axis (led by China -- which must rankle Putin). In this era, Western governments and peoples think Putin's both bad and intolerable. They may also start to feel the same way about Xi, depending on how closely Xi aligns China with Russia.
Expect defence spending and preparedness, in Europe and the USA, to be higher from now on.