Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^\mu [\varphi (u)]=0$. Here $\mathcal{L}^\mu$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $\varphi:\mathbb{R} \to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain $L^1$-contraction and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations.Comment: To appear in "Advances in Mathematics

On distributional solutions of local and nonlocal problems of porous medium type

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of $$\partial_tu-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=g(x,t)\quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $\varphi$ is merely continuous and nondecreasing and $\mathfrak{L}^{\sigma,\mu}$ is the generator of a general symmetric L\'evy process. This means that $\mathfrak{L}^{\sigma,\mu}$ can have both local and nonlocal parts like e.g. $\mathfrak{L}^{\sigma,\mu}=\Delta-(-\Delta)^{\frac12}$. New uniqueness results for bounded distributional solutions of this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for $\mathfrak{L}^{\sigma,\mu}$. Existence and a priori estimates are deduced from a numerical approximation, and energy type estimates are also obtained.Comment: 6 pages. Minor revision. Added details to Step 2 of the proof of Theorem 3.

Asperity deformation during running-in

Asperities loaded in pure rolling against a hard, smooth surface will often be deformed at the first contact event and will thereby experience high normal stress, presumably of a magnitude near the Vickers hardness of the softer material. Continued running-in can be imagined to develop into lower stress levels and an increase of contact area. An asperity model simulating a running-in process of rough surfaces with lengthy protractions in the rolling direction was investigated. After a limited range of only about 104 contact events a state of very low deformation rate was found

Ny vejledning fra Miljøstyrelsen: ”Støj fra veje”: Støjkortlægning og støjhandlingsplaner

Miljøstyrelsen udsendte i juli 2007 den ny støjvejledning, som indfører en ny støjindikator, en ny støjberegningsmetode og nye vejledende støjgrænser. Den ny indikator er den samme, som benyttes til støjkortlægning, og støjgrænsen hedder herefter 58 dB. EU direktiv 2002/49 om støjkortlægning og støjhandlingsplaner blev implementeret i Danmark i 2004, og en revideret bekendtgørelse blev udsendt sammen med en fyldig vejledning i 2006. Fristen for første runde af støjkortlægning er fastsat til 30. juni 2007. Som opfølgning på støjkortlægningen skal de samme myndigheder, der kar kortlagt støjen, udarbejde en handlingsplan om, hvordan man i den forestående 5-års periode vil håndtere støjproblemerne. Mens der er krav for, hvordan støjhandlingsplaner skal udformes, er der ikke krav til deres indhold. Fristen for støjhandlingsplaner er 18. juli 2008. Støjkortlægningen er blevet noget forsinket, men vil i det væsentlige være bragt til ende i løbet af efteråret. Det forventes derfor, at også støjhandlingsplanerne i de enkelte kommuner forsinkes. Præsentationen her gør rede for de ny regler for støj i ”Støj fra veje”, og samspillet mellem støjkortlægning og den ny vejledning. Desuden beskrives de forskellige muligheder, der kan tages op i en støjhandlingsplan med udspring i et inspirationsmateriale ”Styr på støjen”, som Miljøstyrelsen udsendte til alle kommuner i februar 2008

Nonlocal degenerate parabolic hyperbolic equations on bounded domains

We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$\partial_t u+\mathrm{div}\big(f(u)\big)=\mathcal{L}[b(u)]$$on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion operator $\mathcal{L}$ can be any symmetric L{\'e}vy operator (e.g. fractional Laplacians) and $b$ is nondecreasing and allowed to have degenerate regions ($b'=0$). We propose an entropy solution formulation for the problem and show uniqueness and existence of bounded entropy solutions under general assumptions. The uniqueness proof is based on the Kru\v{z}kov doubling of variables technique and incorporates several a priori results derived from our entropy formulation: an $L^\infty$-bound, an energy estimate, strong initial trace, weak boundary traces, and a \textit{nonlocal} boundary condition. The existence proof is based on fixed point iteration for zero-order operators $\mathcal{L}$, and then extended to more general operators through approximations, weak-$\star$ compactness of approximate solutions $u_n$, and \textit{strong} compactness of $b(u_n)$. Strong compactness follows from energy estimates and arguments we introduce to transfer weak regularity from $\partial_t u_n$ to $\partial_t b(u_n)$.Our work can be seen as both extending nonlocal theories from the whole space to domains and local theories on domains to the nonlocal case. Unlike local theories our formulation does not assume energy estimates. They are now a consequence of the formulation, and as opposed to previous nonlocal theories, play an essential role in our arguments. Several results of independent interest are established, including a characterization of the $\mathcal{L}$'s for which the corresponding energy/Sobolev-space compactly embeds into $L^2$

Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$\partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T),$$ where $\mathfrak{L}^{\sigma,\mu}$ is a general symmetric diffusion operator of L\'evy type and $\varphi$ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators $\mathfrak{L}^{\sigma,\mu}$ are the (fractional) Laplacians $\Delta$ and $-(-\Delta)^{\frac\alpha2}$ for $\alpha\in(0,2)$, discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L\'evy operators, allows us to give a unified and compact {\em nonlocal} theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions -- including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in \cite{DTEnJa17b}. We also present some numerical tests, but extensive testing is deferred to the companion paper \cite{DTEnJa18b} along with a more detailed discussion of the numerical methods included in our theory.Comment: 34 pages, 3 figures. To appear in SIAM Journal on Numerical Analysi