论文标题
高斯初始条件的非线性热方程式的标准通货膨胀
Norm inflation for a non-linear heat equation with Gaussian initial conditions
论文作者
论文摘要
我们考虑一个非线性热方程$ \ partial_t u =ΔU + b（u，du） + p（u） + p（u）$，$ d $ d $维圆环，其中$ p $最多是$ 3 $，$ b $的多项式，$ b $是一张双线性地图，不是总衍生品。我们表明，如果初始条件$ U_0 $取自具有指定协方差的一系列平滑高斯字段，则$ u $具有很高的可能性。结果的结果是，没有Banach的分布空间，可以在3D圆环上带有高斯自由场，并且deTurck-yang-mills热流量连续延伸到此，这对arXiv的最新拟合良好导致了ARXIV：2111.10652和ARXIV：2201.03487。另一个结果是（确定性的）非线性热方程式表现出规范通胀，因此在besov空间中的每个点$ b^{ - 1/2} _ {\ infty，\ infty} $;空间$ b^{ - 1/2} _ {\ infty，\ infty} $是一个端点，因为该方程对于$ b^η_ {\ infty，\ infty，\ infty} $的局部良好，对于每个$η> - \ frac12 $。
We consider a non-linear heat equation $\partial_t u = Δu + B(u,Du)+P(u)$ posed on the $d$-dimensional torus, where $P$ is a polynomial of degree at most $3$ and $B$ is a bilinear map that is not a total derivative. We show that, if the initial condition $u_0$ is taken from a sequence of smooth Gaussian fields with a specified covariance, then $u$ exhibits norm inflation with high probability. A consequence of this result is that there exists no Banach space of distributions which carries the Gaussian free field on the 3D torus and to which the DeTurck-Yang-Mills heat flow extends continuously, which complements recent well-posedness results in arXiv:2111.10652 and arXiv:2201.03487. Another consequence is that the (deterministic) non-linear heat equation exhibits norm inflation, and is thus locally ill-posed, at every point in the Besov space $B^{-1/2}_{\infty,\infty}$; the space $B^{-1/2}_{\infty,\infty}$ is an endpoint since the equation is locally well-posed for $B^η_{\infty,\infty}$ for every $η>-\frac12$.
