Results 51 to 60 of about 389 (66)
On two notions of solutions to the obstacle problem for the singular porous medium equation [PDF]
arXiv, 2023 We show that two different notions of solutions to the obstacle problem for the porous medium equation, a potential theoretic notion and a notion based on a variational inequality, coincide for regular enough compactly supported obstacles.
arxiv
Total Variation Flow and Sign Fast Diffusion in one dimension [PDF]
arXiv, 2011 We consider the dynamics of the Total Variation Flow (TVF) $u_t=\div(Du/|Du|)$ and of the Sign Fast Diffusion Equation (SFDE) $u_t=\Delta\sign(u)$ in one spatial dimension. We find the explicit dynamic and sharp asymptotic behaviour for the TVF, and we deduce the one for the SFDE by an explicit correspondence between the two equations.
arxiv
Higher integrability in the obstacle problem for the fast diffusion equation [PDF]
arXiv, 2020 We prove local higher integrability of the spatial gradient for solutions to obstacle problems of porous medium type in the fast diffusion case $m<1$. The result holds for the natural range of exponents that is known from other regularity results for porous medium type equations. We also cover the case of signed solutions.
arxiv
Intrinsic Harnack's inequality for a general nonlinear parabolic equation in non-divergence form [PDF]
arXiv, 2023 We prove the intrinsic Harnack's inequality for a general form of a parabolic equation that generalizes both the standard parabolic $p$-Laplace equation and the normalized version arising from stochastic game theory. We prove each result for the optimal range of exponents and ensure that we get stable constants.
arxiv
Stability for systems of porous medium type [PDF]
arXiv, 2020 We establish stability properties of weak solutions for systems of porous medium type with respect to the exponent $m$. Thereby we treat stability for the local case as well as for Cauchy-Dirichlet problems. Both degenerate and singular cases are covered.
arxiv
Stagnation, Creation, Breaking [PDF]
arXiv, 2014 This note deals with the mono-dimensional equation: $\d_t u -\d_x L(u_x) =f$ with $L(\cdot)$ merely monotone. The goal is to examine the features of facets -- flat regions of graphs of solutions appearing as $L(\cdot)$ suffers jumps. We concentrate on examples explaining strong stability, creation and even breaking of facets.
arxiv
An Introduction to Barenblatt Solutions for Anisotropic $p$-Laplace Equations [PDF]
arXiv, 2020 We introduce Fundamental solutions of Barenblatt type for the equation $u_t=\sum_{i=1}^N \bigg( |u_{x_i}|^{p_i-2}u_{x_i} \bigg)_{x_i}$, $p_i >2 \quad \forall i=1,..,N$, on $\Sigma_T=\mathbb{R}^N \times[0,T]$, and we prove their importance for the regularity properties of the solutions.
arxiv
arXiv, 2014 We study a singular one-dimensional parabolic problem with initial data in the $BV$ space, the energy space, for various boundary data. We pay special attention to Dirichlet conditions, which need not satisfied in a pointwise manner. We study the facet creation process and the extinction of solutions caused by the evolution of facets. Our major tool is
arxiv
Dirichlet boundary conditions for degenerate and singular nonlinear parabolic equations [PDF]
arXiv, 2014 We study existence and uniqueness of solutions to a class of nonlinear degenerate parabolic equations, in bounded domains. We show that there exists a unique solution which satisfies possibly inhomogeneous Dirichlet boundary conditions. To this purpose some barrier functions are properly introduced and used.
arxiv
A level set crystalline mean curvature flow of surfaces [PDF]
arXiv, 2016 We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by regularized problems, and we also show the uniqueness and existence of a level set flow for bounded crystals.
arxiv