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Global Existence for a Fourth-Order Parabolic Equation With Couette Flow in the Whole Space
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DOI：10.1002/mma.70723.png)
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Abstract
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We study the global existence and long-time decay of solutions to fourth-order parabolic equations on under background Couette flows. Using the Green's function method, we show that sufficiently strong Couette flows induce enhanced dissipation that suppresses finite-time blow-up. Due to the lack of an explicit representation of the associated Green's function, we first derive sharp estimates for its Fourier transform and then transfer them to obtain bounds in physical space. Moreover, for equations involving both second- and fourth-order diffusion, we prove that, whenever global solutions exist, their long-time decay is governed by the second-order diffusion. Consequently, the decay rate coincides with that of the pure second-order case and is faster than that of equations involving only fourth-order diffusion.
Keywords:
Couette flow
dissipation-enhancing mechanism
fourth-order parabolic
Green's function
suppression of blow up
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1.8
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605
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