The nonlinear Benjamin-Ono equation on the torus is approximately controllable in L2 with 2D input, implying almost-sure unbounded Sobolev growth for nondegenerate random forcing.
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The Benjamin-Ono equation with 2D control input: approximate controllability and its application
The nonlinear Benjamin-Ono equation on the torus is approximately controllable in L2 with 2D input, implying almost-sure unbounded Sobolev growth for nondegenerate random forcing.