Large time decay for the magnetohydrodynamics system in \(\dot{H}^s (\mathbb{R}^n)\). (English) Zbl 1447.35064
Summary: We show that \(t^{s/2} \| (\boldsymbol{u}, \boldsymbol{b})(t) \|_{\dot{H}^s(\mathbb{R}^n)} \to 0\) as \(t \to \infty\) for global Leray solutions \((\boldsymbol{u}, \boldsymbol{b})(t)\) of the incompressible MHD equations, where \(2 \leq n \leq 4\) and \(s \geq 0\) (real). We also provide some related results and, as a consequence, the following general decay property:
\[
\lim_{t \to \infty} t^{\gamma(n,m,q)} \bigl\| \bigl(D^m \boldsymbol{u}, D^m \boldsymbol{b}\bigr) (t) \bigr\|_{\mathbf{L}^q(\mathbb{R}^n)} = 0.
\]
Where \(\gamma(n,m,q) = \frac{n}{4} + \frac{m}{2} - \frac{n}{2q}\), for each \(2 \leq q \leq \infty, n=2,3,4\) and \(m \geq 0\) integer.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
Keywords:
decay rates in homogeneous Sobolev spaces; MHD incompressible; time decay of derivatives; long time behavior; \(L^q\) estimatesReferences:
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