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Large Time Decay for the Magnetohydrodynamics System in H s ˙ ( R n ) $\dot{\boldsymbol{H}^{s}}(\mathbb{R}^{n})$
Acta Applicandae Mathematicae ( IF 1.2 ) Pub Date : 2019-07-29 , DOI: 10.1007/s10440-019-00276-y
Wilberclay G. Melo , Cilon F. Perusato , Robert H. Guterres , Juliana R. Nunes

We show that \(t^{s/2} \| (\boldsymbol{u},\boldsymbol{b})(t) \|_{{\dot{H}^{s}(\mathbb{R}^{n})}} \to0 \) as \(t \to\infty\) for global Leray solutions \((\boldsymbol{u},\boldsymbol{b})(t)\) of the incompressible MHD equations, where \(2 \leq n \leq4\) and \(s \geq0\) (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 \(m\geq0 \) integer.

中文翻译:

H s(R n)$ \ dot {\ boldsymbol {H} ^ {s}}(\ mathbb {R} ^ {n})$中磁流体动力学系统的大时衰减

我们证明\(t ^ {s / 2} \ |(\ boldsymbol {u},\ boldsymbol {b})(t)\ | _ {{\ dot {H} ^ {s}(\ mathbb {R} ^ {n})}} \ to0 \)作为不可压缩MHD方程的全局Leray解\((\ boldsymbol {u},\ boldsymbol {b})(t)\)\(t \ to \ infty \),其中\(2 \ leq n \ leq4 \)\(s \ geq0 \)(真实)。我们还提供了一些相关结果,并因此提供了以下一般衰减属性:$$ \ lim_ {t \ to \ infty} t ^ {\ gamma(n,m,q)} \ bigl \ | \ bigl(D ^ {m} \ boldsymbol {u},D ^ {m} \ boldsymbol {b} \ bigr)(t)\ bigr \ | _ {\ mathbf {L} ^ {q}(\ mathbb {R} ^ {n})} =0。$$其中\(\ gamma(n,m,q)= \ frac {n} {4} + \ frac {m} {2}-\ frac {n} {2q} \),对于每个\(2 \ leq q \ leq \ infty \)\(n = 2,3,4 \)\(m \ geq0 \)整数。
更新日期:2019-07-29
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