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Well- and ill-posedness for a class of the 3D-generalized Kuramoto–Sivashinsky equations

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Abstract

For a general domain \(\Omega \subset {\mathbb {R}}^3\), we examine the initial value problem (IVP) and final value problem (FVP) for a class of the generalized Kuramoto–Sivashinsky equations (GKSEs). By considering the IVP of GKSE, we prove the global well-posedness theory (local existence, continuation and global existence or finite-time blow-up) in \(W^{1,p}(\Omega )\) for \(p\ge 2\). For studying the FVP of GKSE, the problem is severely ill-posed. We suggest the Fourier truncation method to regularize the problem. The well-posedness properties (existence and regularity) in the space \(W^{1,p}(\Omega )\) of the regularized solution are proved. Moreover, the error estimate in \(W^{1,p}(\Omega ), p\ge 2\) is represented to verify the convergence of the regularization method when the sought solution belongs to abstract Gevrey spaces.

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Acknowledgements

The author was supported by the Van Lang University.

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  1. Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam

    Vo Van Au

  2. Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam

    Vo Van Au

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Au, V.V. Well- and ill-posedness for a class of the 3D-generalized Kuramoto–Sivashinsky equations. Z. Angew. Math. Phys. 74, 247 (2023). https://doi.org/10.1007/s00033-023-02145-z

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