We show that if \(\mathbf {u}\) is a weak solution to the Navier–Stokes initial–boundary value problem with Navier’s slip boundary conditions in \(Q_T:=\Omega \times (0,T)\), where \(\Omega \) is a domain in \({{\mathbb {R}}}^3\), then an associated pressure p exists as a distribution with a certain structure. Furthermore, we also show that if \(\Omega \) is a “smooth” domain in \({{\mathbb {R}}}^3\) then the pressure is represented by a function in \(Q_T\) with a certain rate of integrability. Finally, we study the regularity of the pressure in sub-domains of \(Q_T\), where \(\mathbf {u}\) satisfies Serrin’s integrability conditions.

中文翻译：

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带有Navier边界条件的Navier–Stokes方程的弱解的压力

我们证明，如果\（\ mathbf {u} \）是在\（Q_T：= \ Omega \ times（0，T）\）中具有Navier滑移边界条件的Navier-Stokes初始边界值问题的弱解决方案，其中\（\ Omega \）是\（{{\ mathbb {R}}} ^ 3 \）中的一个域，则关联压力p作为具有特定结构的分布而存在。此外，我们还表明，如果\（\ Omega \）是\（{{\ mathbb {R}}} ^ 3 \）中的“平滑”域，则压力由\（Q_T \）中的函数表示，一定的集成度。最后，我们研究\（Q_T \）子域中压力的规则性，其中\（\ mathbf {u} \） 满足Serrin的可集成性条件。