What will happen to some specific anime if the studio who made those animes has gone bankrupt?

In US law, apparently there isn't a supreme court ruling yet. Generally, the legal opinions I've read (not being a corporate lawyer myself) seem to state that it is up to the trustee appointed to handle things. If they decide the existing licence agreement adds value, they can decide to take up the agreement, otherwise they may decide not to, and sell the license at auction or offer it to another company. If a trustee is not appointed (Debtor in Possession) the agreement remains, with some provisos. This is based on US bancruptcy law (section 11) which says sometion along the lines of: the trustee, subject to the court’s approval, may assume or reject any executory contract or unexpired lease of the debtor. It includes pretty much all intellectual property licences, with the exception of Trademarks which are handled differently. The person/company who had the license can file suit for damages if they lose it. This seems to be somewhat similar to Swiss and German law too. Wasn't able to find a translation I could read on Japanese law. I would assume that would be more pertinent. As there has been no announcements that Crunchyroll is removing them, there hopefully isn't going to be an issue.

References:

• 11 U.S.C. § 362.
• 11 U.S.C. § 365.
• Marcelo Halpern, Bankruptcy Issues in Intellectual Property Licensing, in 2 Understanding the Intellectual Property License 2012, at 245, 271 (Practising Law Inst. 2012); Riback, supra note 12, at 641.
• Vern Countryman, Executory Contracts in Bankruptcy Law: Part I, 57 Minn. L. Rev. 439, 460 (1973).
• Stuart M. Riback, Intellectual Property Licenses: The Impact of Bankruptcy, in Understanding the Intellectual Property License 2006, at 607, 617 (Practising Law Inst. 2006).
• Stuart S. Moskowitz; Intellectual Property Licenses in Bankruptcy: New "Veto Power" for Licensees Under Section 365(n); The Business Lawyer Vol. 44, No. 3 (May 1989), pp. 771-790
• Peter N. Brown, Morse, Barnes-Brown, Pendleton PC, Licensing Intellectual Property from and to Distressed Companies (2005), www.mbbp.com/resources/iptech/pdfs/ip_distressed.pdf.
• https://www.americanbar.org/groups/intellectual_property_law/publications/landslide/2013-14/september-october-2013/intellectual-property-licenses-bankruptcy
• https://www.mondaq.com/unitedstates/licensing-syndication/677428/treatment-of-ip-licenses-in-bankruptcy

Note: I put in the first few references on my list somewhat under protest, as I don't think they add value. As I said before, I'd be more interested in the Japanese law. Also, I collated the answer from skimming through about 50 different legal opinions a while back, for a piece of fiction was editing. I'm not typing all of that into this

TL;DR: the anime studio doesn't own the complete rights to the anime, so in most cases there will be no problem in the case the studio goes under.

Long version:

The owner of the rights to an anime in modern times is almost always a 制作委員会, which I usually translate as "production partnership".

This is a partnership that is set up in order to fund the anime, while splitting the risk with other sponsors. This includes companies like the publisher of the original work, the music company, figure companies, advertising companies, TV stations, etc. This may or may not include the anime production studio itself. Either way, in the case the studio ceases to exist, this would not affect any agreement with foreign publishers, as the rights are still being managed by the production partnership.

It should be noted that just because a company goes bankrupt doesn't necessarily mean it loses all capability to create anime (depending on the type of bankruptcy). There have been several cases like this, for example Tatsunoko Production. Mangrove, whose anime you have mentioned, has ceased to exist as of 2016, but there has been no removal of the anime (and Mangrove has never opted into the partnership, so they never had any rights to the anime itself anyway). It does seem that Mangrove's bankruptcy affected Gangsta's BD release, but I would assume that had more to do with the common practice for anime studios to brush up and fix TV anime for the disk releases.

There have been cases where studios go bankrupt and production partnerships enlisted a different studio to produce the sequel. Strike Witches and Saki are examples of this (in the case of Saki, Studio Gokumi is a studio founded by former Gonzo employees).

Sources (all in Japanese):

• Structure of production partnerships

  1. Naver
  2. Wiki
  3. Netolabo

• Anime (includes production studio and production partnership participants)

  4. Hayate no Gotoku series
  5. Saki
  6. Strike Witches

• Studios

  7. Studio Gokumi
  8. Mangrove
  9. Gonzo

• Bankruptcy-related (I can't find the official statements, if there were any made public)

  10. Mangrove article
  11. List of anime studios that went bankrupt