Let $R$ be a complete DVR with fraction field $K$, $X$ be a regular scheme flat over $R$. Let $L$ be a finite field extension of $K$ and $Q$ be the integral closure of $R$ in $L$. Denote by $Y:=X \times_R Q$ the base change of $X$. Is $Y$ a regular scheme? If not true in general, is there any additional assumption on $R$ for which this holds true?
PS. One can assume that the characteristic of $K$ is zero.