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Juhana Siljander

Publications and source records attributed to Juhana Siljander.

11 recordsLinked to original sources
arXiv2018

On the parabolic Harnack inequality for non-local diffusion equations

We settle the open question concerning the Harnack inequality for globally positive solutions to non-local in time diffusion equations by constructing a counter-example for dimensions $d\geβ$, where $β\in(0,2]$ is the order of the equation with respect to the spatial variable. The equation can be non-local both in time and in space but for the counter-example it is important that the equation has a fractional time derivative. In this case, the fundamental solution is singular at the origin for all times $t>0$ in dimensions $d\geβ$. This underlines the markedly different behavior of time-fractional diffusion compared to the purely space-fractional case, where a local Harnack inequality is known. The key observation is that the memory strongly affects the estimates. In particular, if the initial data $u_0 \in L^q_{loc}$ for $q$ larger than the critical value $\tfrac dβ$ of the elliptic operator $(-Δ)^{β/2}$, a non-local version of the Harnack inequality is still valid as we show. We also observe the critical dimension phenomenon already known from other contexts: the diffusion behavior is substantially different in higher dimensions than $d=1$ provided $β>1$, since we prove that the local Harnack inequality holds if $d<β$.

math.AP↗
arXiv2018

Boundary regularity for the porous medium equation

We study the boundary regularity of solutions to the porous medium equation $u_t = Δu^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the parabolic boundary has a solution which attains the boundary values, provided that the spatial domain satisfies the elliptic Wiener criterion. This condition is known to be optimal, and it is a consequence of our main theorem which establishes a barrier characterization of regular boundary points for general -- not necessarily cylindrical -- domains in ${\bf R}^{n+1}$. One of our fundamental tools is a new strict comparison principle between sub- and superparabolic functions, which makes it essential for us to study both nonstrict and strict Perron solutions to be able to develop a fruitful boundary regularity theory. Several other comparison principles and pasting lemmas are also obtained. In the process we obtain a rather complete picture of the relation between sub/super\-para\-bolic functions and weak sub/super\-solu\-tions.

math.AP↗
arXiv2016

Sharp Regularity for Weak Solutions to the Porous Medium Equation

Let $u$ be a nonnegative, local, weak solution to the porous medium equation for $m\ge2$ in a space-time cylinder $Ω_T$. Fix a point $(x_o,t_o)\inΩ_T$: if the average \[ a{\buildrel\mbox{def}\over{=}}\frac1{|B_r(x_o)|}\int_{B_r(x_o)}u(x,t_o)\,dx>0, \] then the quantity $|\nabla u^{m-1}|$ is locally bounded in a proper cylinder, whose center lies at time $t_o+a^{1-m}r^2$. This implies that in the same cylinder the solution $u$ is Hölder continuous with exponent $α=\frac1{m-1}$, which is known to be optimal. Moreover, $u$ presents a sort of instantaneous regularisation, which we quantify.

math.AP↗
arXiv2016

On the interior regularity of weak solutions to the 2-D incompressible Euler equations

We consider the 2-D incompressible Euler equations in a bounded domain and show that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity result $$u \in L_{\rm loc}^{2+\varepsilon}(Ω_T) \implies {\rm regularity}$$ for weak solutions in the energy space $L_t^\infty L_x^2$ satisfying appropriate vorticity estimates. The argument is completely local in nature as the result follows from the structural properties of the equation alone, while completely avoiding all sorts of boundary conditions and related gradient estimates. To the best of our knowledge, the approach we follow is new in the context of Euler equations and provides an alternative look at interior regularity issues. We also show how our method can be used to give a modified proof of the classical Serrin condition for the regularity of the Navier-Stokes equations in any dimension.

math.AP↗
arXiv2015

Representation of solutions and large-time behavior for fully nonlocal diffusion equations

We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems: (i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the solution tends to the fundamental solution, (iii) optimal $L^2$-decay of mild solutions in all dimensions, (iv) $L^2$-decay of weak solutions via energy methods. The first result relies on a delicate analysis of the definition of classical solutions. After proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of (ii). Next we use Fourier analysis techniques to obtain the optimal decay rate for mild solutions. Here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions. Consequently, the decay rate does not anymore improve when the dimension increases. The theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis. Finally, we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation. Our main idea is to show that the $L^2$-norm is actually a subsolution to a purely time-fractional problem which allows us to use the known theory to obtain the result.

math.AP↗
arXiv2014

Everywhere differentiability of viscosity solutions to a class of Aronsson's equations

For any open set $Ω\subset\mathbb R^n$ and $n\ge 2$, we establish everywhere differentiability of viscosity solutions to the Aronsson equation $$ =0 \quad \rm in\ \ Ω, $$ where $H$ is given by $$H(x,\,p)= =\sum_{i,\,j=1}^na^{ij}(x)p_i p_j,\ x\inΩ, \ p\in\mathbb R^n, $$ and $A=(a^{ij}(x))\in C^{1,1}(\barΩ,\mathbb R^{n\times n})$ is uniformly elliptic. This extends an earlier theorem by Evans and Smart \cite{es11a} on infinity harmonic functions.

math.AP↗
arXiv2014

Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\mathbb{R}^d$

We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in $\mathbb{R}^d$. An important special case is the time-fractional diffusion equation, which has seen much interest during the last years, mostly due to its applications in the modeling of anomalous diffusion processes. We follow three different approaches and techniques to study this particular case: (A) estimates based on the fundamental solution and Young's inequality, (B) Fourier multiplier methods, and (C) the energy method. It turns out that the decay behaviour is markedly different from the heat equation case, in particular there occurs a {\em critical dimension phenomenon}. The general subdiffusion case is treated by method (B) and relies on a careful estimation of the underlying relaxation function. Several examples of kernels, including the ultraslow diffusion case, illustrate our results.

math.AP↗
arXiv2011

Hölder continuity for Trudinger's equation in measure spaces

We complete the study of the regularity for Trudinger's equation by proving that weak solutions are Hölder continuous also in the singular case. The setting is that of a measure space with a doubling non-trivial Borel measure supporting a Poincaré inequality. The proof uses the Harnack inequality and intrinsic scaling.

math.AP↗
arXiv2010

Local Hölder continuity for doubly nonlinear parabolic equations

We give a proof of the Hölder continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincaré inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods.

math.AP↗