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$L^2$ norm for solutions of evolution equations driven by different elliptic operators

Not necessarily. I mean, it depends upon the torus you consider. Notice that in the case of the standard one ${\mathbb T}^d={\mathbb R}^d/{\mathbb Z}^d$, the answer is positive. But if you torus is ${\...
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Change of variables for obtaining a unitary group

Let $u=u_1+iu_2$, $v=v_1+iv_2$ with $u_1,u_2,v_1,v_2$ being real valued. What one would like to do is find a $2\times 2$ matrix of real symmetric operators $$ \begin{pmatrix} A & B\\ C & D \...
2 votes

Change of variables for obtaining a unitary group

The problem is to show that the equation $$ iu_t+\Delta u-2\,{\rm Re}\,u=F(u),\label{1}\tag{1} $$ with $u=Vv$, is equivalent to $$i v_t - \sqrt{- \Delta (2-\Delta)} v = - i V^{-1} i F(V v).\label{2}\...

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