抛物型PDE的螺线管Lipschitz截断
Solenoidal Lipschitz truncation for parabolic PDE's
摘要：
我们考虑函数$u\in L^\infty(L^2)\cap L^p(W^{1,p})$与$1<p<\infty$的时间空间域上。非线性演化偏微分方程的解通常属于这些空间。许多应用需要$u$的Lipschitz近似$u_λ$，其在大集合上与$u$一致。对于流体力学中出现的问题，需要使用螺线管（无发散）函数。因此，我们构造了一个Lipschitz近似，它也是螺线管。作为应用，我们修正了文献[DRW 10]中关于非定常广义牛顿流体存在性的证明。由于${\rm div} u_λ=0$，我们能够在无压力的配方，大大简化了证明工作。我们还提供了一个简化的方法来固定螺线管Lipschitz截断[BDF 12]。
英文摘要：
We consider functions $u\in L^\infty(L^2)\cap L^p(W^{1,p})$ with $1<p<\infty$ on a time space domain. Solutions to non-linear evolutionary PDE's typically belong to these spaces. Many applications require a Lipschitz approximation $u_λ$ of $u$ which coincides with $u$ on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids in [DRW10]. Since ${\rm div} u_λ=0$, we are able to work in the pressure free formulation, which heavily simplifies the proof. We also provide a simplified approach to the stationary solenoidal Lipschitz truncation of [BDF12].
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