论文标题
全球存在用于准论不可压缩的Navier-Stokes模型的解决方案
The Global Existence of Solutions to a Quasi-Relativistic Incompressible Navier-Stokes Model
论文作者
论文摘要
我们通过在普通Navier-Stokes方程中修改对流术语来以$ 3 $尺寸引入新的修改后的Navier-Stokes模型。这是通过替换对流术语$（\ textbf {u} \ cdot \ nabla）\ textbf {u} $的$（\ textbf {v} \ cdot \ nabla）\ textbf {u} $而完成的。 $ \ textbf {v} = c \ textbf {u}/\ sqrt {c^2+| \ textbf {u} |^2} $，其中$ c $是光的速度。因此，我们有$ | \ textbf {v} | \ leq c $，对于$ | \ textbf {u} | \ ll c $，我们有$ \ textbf {v} \ oft textbf {u} $。因此，该系统的解决方案应与在物理合理条件下的普通Navier-Stokes方程的解决方案产生良好的近似值。对流术语的修改是在\ cite {jaraczlee}中完成的工作的自然发展。 $ | \ textbf {v} | \ leq c $的属性体现了相对性问题不能比光速更快的概念，从而为模型提供了其名称。我们证明存在一个强大的解决方案$ \ textbf {u} \ in l^2（0，t; \ textbf {h}^2）\ cap l^{\ infty}（0，t; \ \ \ \ textbf {v}）$ with $ \ textbf {u} equations on either a smooth bounded domain $U\subset \textbf{R}^3$ or the flat $3$-torus $\mathbb{T}$ for any initial velocity $\textbf{u}_0 \in \textbf{V}$ and any forcing function $\textbf{f}\in L^2(0, T; \ textbf {l}^2）$。对于数据的较小，没有任何假设。这里$ \ textbf {v} $是弱分歧的空间，其中$ h^1 $的组件在边界上消失了。我们还证明了这种强大解决方案的独特性。尽管我们的修改有些临时，但它表明，尽管更复杂，但与普通的Navier-Stokes方程相比，包含特殊和一般相对性方面的方程可能具有更好的存在和独特性。
We introduce a new modified Navier-Stokes model in $3$ dimensions by modifying the convection term in the ordinary Navier-Stokes equations. This is done by replacing the convective term $(\textbf{u}\cdot \nabla) \textbf{u}$ by $(\textbf{v}\cdot \nabla)\textbf{u}$ with $\textbf{v}=c\textbf{u}/\sqrt{c^2+|\textbf{u}|^2}$ where $c$ is the speed of light. Thus we have that $|\textbf{v}|\leq c$ and for $|\textbf{u}|\ll c$ we have $\textbf{v} \approx \textbf{u}$. Thus the solutions to this system should yield a good approximation to the solutions of the ordinary Navier-Stokes equations under physically reasonable conditions. The modification of the convective term is a natural progression of the work done in \cite{JaraczLee}. The property that $|\textbf{v}|\leq c$ embodies the notion that in relativity matter can't travel faster than the speed of light, giving the model its name. We prove that there exists a strong solution $\textbf{u} \in L^2(0, T; \textbf{H}^2) \cap L^{\infty}(0, T; \textbf{V})$ with $\textbf{u}' \in L^2(0, T; \textbf{L}^2)$ to our system of equations on either a smooth bounded domain $U\subset \textbf{R}^3$ or the flat $3$-torus $\mathbb{T}$ for any initial velocity $\textbf{u}_0 \in \textbf{V}$ and any forcing function $\textbf{f}\in L^2(0, T; \textbf{L}^2)$. No assumption on the smallness of the data is necessary. Here $\textbf{V}$ is the space of weakly divergence free vector fields with components in $H^1$ which vanish on the boundary. We also prove the uniqueness of this strong solution. Though our modification is somewhat ad-hoc, it suggests that though more complicated, equations incorporating aspects of special and general relativity might have better existence and uniqueness properties than the ordinary Navier-Stokes equations.
