Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces
Lower Bounds for an Integral Involving Fractional Laplacians and the Generalized Navier-Stokes Equations in Besov Spaces
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DOI：
10.1007/s00220-005-1483-6
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发表时间：
2006-05
影响因子：
2.4
通讯作者：
Jiahong Wu
中科院分区：
文献类型：
--
作者：
Jiahong Wu
在Lp相关空间中估计耗散型偏微分方程的解时，常常需要包含耗散项的积分的下界。如果耗散项由通常的拉普拉斯-Δ给出，则可以通过分部积分和嵌入不等式导出下界。然而，当拉普拉斯算子被分数拉普拉斯算子（−Δ）α取代时，分部积分法不再适用。本文将（-Δ）α的逐点不等式与分数阶导数的伯恩斯坦不等式相结合，得到了（-Δ）α积分的下界.作为这些下界的应用，我们建立了Besov空间中广义Navier-Stokes方程解的存在唯一性。广义Navier-Stokes方程是将Navier-Stokes方程中的−Δ替换为（−Δ）α的方程。
When estimating solutions of dissipative partial differential equations inLp-related spaces, we often need lower bounds for an integral involving the dissipative term. If the dissipative term is given by the usual Laplacian −Δ, lower bounds can be derived through integration by parts and embedding inequalities. However, when the Laplacian is replaced by the fractional Laplacian (−Δ)α, the approach of integration by parts no longer applies. In this paper, we obtain lower bounds for the integral involving (−Δ)αby combining pointwise inequalities for (−Δ)αwith Bernstein's inequalities for fractional derivatives. As an application of these lower bounds, we establish the existence and uniqueness of solutions to the generalized Navier-Stokes equations in Besov spaces. The generalized Navier-Stokes equations are the equations resulting from replacing −Δ in the Navier-Stokes equations by (−Δ)α.