Can PhD students study in universities in Germany, in engineering courses, without knowing the German language? Can they research and live without any problems?
Does applying to these universities require knowledge of the German language?
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I know programs in other fields where the official graduate school language is English (e.g., my undergraduate alma mater - over 20 years back even), but am not personally familiar with engineering in Germany. You need to check the online program description for the universities you're interested in. They certainly exist - here is one example. The entire page is in English. It's an interview with international students about their experiences in the program, so it might be helpful, or at least interesting as well.
I had an Israeli professor who didn't speak German. He was tenured, taught in English even at the undergraduate level, and still managed to be probably the most popular teacher. While Germans generally speak some English, you just face a somewhat restricted social life, and should learn German while in the program to enjoy your time in a PhD more.