Discontinuous solutions of Hamilton-Jacobi equations versus Radon measure-valued solutions of scalar conservation laws: Disappearance of singularities

带跳的非线性Hamilton-Jacobi方程的带跳梯度流

Gradient flows with jumps associated with nonlinear Hamilton-Jacobi equations with jumps

我们分析了由有限集完备向量场集在对合中以Radon测度$u\in \mathcal{U}_a$为容许扰动所产生的带跳跃的梯度流。利用与流动和Hamilton-jacobi方程相关联的相应梯度表示描述了有界梯度流$\{x^u(t,ł)\in B(x^*,3\g)\subseteq \mbn: \,t\in[0,T],\,ł\in B(x^*,2\g)\}$的演化和积分方程$x^u(t,ł)=x\in B(x^*,\g), \,t\in[0,T]$的唯一解$ł=ψ^u(t,x)\in B(x^*,2\g)\subseteq \mbn$.

具有吸收项的Hamilton-Jacobi抛物型方程解的初迹

Initial trace of solutions of Hamilton-Jacobi parabolic equation with absorption

本文研究了方程\[ u\_{t}-Δu+|\nabla u|^{q}=0 \]在$Q\_{Ω,T}=Ω\times\left( 0,T\right) ,$ $T\leqq\infty,$中的非负解的初迹问题，其中$q>0,$和$Ω=\mathbb{R}^{N},$或$Ω$是$\mathbb{R}^{N}$和$u=0$在$\partialΩ\times\left( 0,T\right) .$上的光滑有界区域。我们可以将$t=0$处的迹定义为非负Borel测度$(\mathcal{S} ,u\_{0}),$，其中$S$是无穷大的闭集，和$u\_{0}$是$Ω\backslash\mathcal{S}.$上的Radon测度我们证明了当$q\leqq1.$时迹是Radon测度对于$q\in(1,(N+2)/(N+1)$和任何给定的Borel测度，我们证明了极小解的存在性，当$\mathcal{S}$ $=\overlineω\capΩ$和$ω$是$Ω,$的开子集时，存在性扩展到任何$q\leqq2$当$u\_{0}\in L\_{loc}^{1}(Ω)$和任何$q>1$当$u\_{0}=0$。特别地，对于固定的$t.$，存在一个具有迹$(\mathbb{R}^{N+},0),$的自相似非径向解，其增长率为$\left\vert x\right\vert ^{q^{\prime}}$阶和$\…

Quasi-Invariance of Gaussian Measures Transported by the Cubic NLS with Third-Order Dispersion on $\mathbf{T}$

We consider the Nonlinear Schrödinger (NLS) equation and prove that the Gaussian measure with covariance $(1-\partial_x^2)^{-α}$ on $L^2(\mathbf T)$ is quasi-invariant for the associated flow for $α>1/2$. This is sharp and improves a previous result obtained in \cite{OTT} where the values $α>3/4$ were obtained. Also, our method is completely different and simpler, it is based on an explicit formula for the Radon-Nikodym derivative. We obtain an explicit formula for this latter in the same spirit as in \cite{Cruz1} and \cite{Cruz2}. The arguments are general and can be used to other Hamiltonian…

具有吸收项的Hamilton-Jacobi抛物型方程解的局部和整体估计

Local and global estimates of solutions of Hamilton-Jacobi parabolic equation with absorption

我们得到了方程\[ u_{t}-Δu+|\nabla u|^{q}=0 \]在$Q_{Ω,T}=Ω\times\left( 0,T\right) ,$ $T\leqq\infty,$中的非负解的新的先验估计，其中$q>0,$和$Ω=\mathbb{R}^{N},$或$Ω$是$\mathbb{R}^{N}$和$u=0$在$\partialΩ\times\left( 0,T\right) .$上的光滑有界区域。在$Ω=\mathbb{R}^{N},$的情况下，我们证明了方程的任何解$u\in C^{2,1}(Q_{\mathbb{R}^{N},T})$ 1.1在$Q_{\mathbb{R}^{N} ,T}$中，特别是任何弱解，如果$q\leqq2),$不带条件作为$\left\vert x\right\vert \rightarrow\infty,$满足普适估计\[ \left\vert \nabla u(.,t)\right\vert ^{q}\leqq\frac{1}{q-1}\frac{u(.,t)}% {t},\qquad\text{in }Q_{\mathbb{R}^{N},T}. \]，并且我们证明了$u$的增长性是由$C(t+t^{-1/(q-1}% )(1+\left\vert x\right\vert ^{q^{\prime}}),$限制的，其中$C$依赖于$u.$.…

具有一般凸通量的标量守恒律的最小熵条件

Minimal Entropy Conditions for Scalar Conservation Laws with General Convex Fluxes

本文研究一般凸通量下一维标量守恒律的最小熵条件，证明单个严格凸熵对即可在 $L^\infty_{\rm loc}$ 中唯一确定熵解，并推广至 $L^p_{\rm loc}$。