Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term

EnVarA-FEM for the flux-limited porous medium equation

Developmental Biology studies the processes whereby animals and plants grow and develop. One of the most studied scenarios in this discipline is the development of the neural tube upon Sonic Hedgehog signaling. We review several mathematical approaches that have been used to describe the former system in a quantitative way, paying attention to both their merits and their drawbacks. Experimental results suggest that transient dynamics can be quite important for the understanding of the patterning process. It has not been possible to fully replicate these dynamics by means of the available models yet, an important problem being to deal with the finite propagation speed of signaling interfaces. While continuous models relying on finite propagation speed mechanisms have been advocated elsewhere, we discuss here a spatially discrete multiscale model allowing to cope with this problem.

Laplace operators on metric graphs give rise to Klein–Gordon and wave operators. Solutions of the Klein–Gordon equation and the wave equation are studied and finite propagation speed is established. Massive, free quantum fields are then constructed, whose commutator function is just the Klein–Gordon kernel. As a consequence of finite propagation speed, Einstein causality (local commutativity) holds. Comparison is made with an alternative construction of free fields involving RT-algebras.

The finite speed of propagation (FSP) was established for certain materials in the 70’s by the American school (Gurtin, Dafermos, Nohel, etc.) for the special case of the presence of memory effects. A different approach can be applied by the construction of suitable super and sub-solutions (Crandall, Nohel, Diaz and Gomez, etc.). In this paper we present an alternative method to prove (FSP) which only uses some energy estimates and without any information coming from the characteristics analysis. The waiting time property is proved for the first time in the literature for this class of nonlocal equations.

It is known that the diffusion equation used to model transport in a large variety of chromatographic techniques has an infinite speed of signal propagation, i.e., if c(x,t) is the concentration at time t, then c(x,t) > 0 for any t > 0. We generalize and solve the telegraph equation, which is known to have a finite speed of signal propagation, to allow for asymmetric convection, as is appropriate for the theory of chromatographic processes. We derive the telegraph equation from a continuous time random walk picture and examine two sources of convection, an asymmetry in sojourn times in states in which diffusing particles move in one direction or the other, and a corresponding asymmetry in the velocities.

This work deals with the study of a nonlinear degenerate pseudoparabolic problem. Arising from the modelling of sedimentary basin formation, the equation degenerates in order to take implicitly into account a constraint on the time derivative of the unknown. An existence result of a solution with an adapted compactness result and qualitative properties of the solutions are proposed (localization, finite speed of propagation, etc.).

The Cauchy problem for first order system $L(t, x, \D_t, \D_x)$ is known to be well posed in $L^2$ when a it admits a microlocal symmetrizer $S(t,x, \xi)$ which is smooth in $\xi$ and Lipschitz continuous in $(t, x)$. This paper contains three main results. First we show that a Lipsshitz smoothness globally in $(t,x, \xi)$ is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of \emph{full symmetrizers} having the same smoothness. This notion was first introduced in \cite{FriLa1}. This is the key point to prove the third result that the existence of microlocal symmetrizer is preserved if one changes the direction of time, implying local uniqueness and finite speed of propagation.

We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in [Formula: see text]. In terms of the enthalpy [Formula: see text], the evolution equation reads [Formula: see text], while the temperature is defined as [Formula: see text] for some constant [Formula: see text] called the latent heat, and [Formula: see text] stands for the fractional Laplacian with exponent [Formula: see text]. We prove the existence of a continuous and bounded selfsimilar solution of the form [Formula: see text] which exhibits a free boundary at the change-of-phase level [Formula: see text]. This level is located at the line (called the free boundary) [Formula: see text] for some [Formula: see text]. The construction is done in 1D, and its extension to [Formula: see text]-dimensional space is shown. We also provide well-posedness and basic properties of very weak solutions for general bounded data [Formula: see text] in several dimensions. The temperatures [Formula: see text] of these solutions are continuous functions that have finite speed of propagation, with possible free boundaries. We obtain estimates on the growth in time of the support of [Formula: see text] for solutions with compactly supported initial temperatures. Besides, we show the property of conservation of positivity for [Formula: see text] so that the support never recedes. On the contrary, the enthalpy [Formula: see text] has infinite speed of propagation and we obtain precise estimates on the tail. The limits [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are also explored, and we find interesting connections with well-studied diffusion problems. Finally, we propose convergent monotone finite-difference schemes and include numerical experiments aimed at illustrating some of the obtained theoretical results, as well as other interesting phenomena.

In this paper we study the speed of propagation of the free boundary of the dam problem. We prove that the free boundary of the saturated part (X = 1) has a finite speed of propagation, which implies that the speed of propagation of the pressure is finite when the medium is not saturated. When the medium is saturated the speed of propagation of the pressure can be infinite.

> 0 is a constant. This equation, which is parabolic, has a very unpleasant feature: a thermal disturbance at any point in the body is felt instantly at every other point; or in terms more suggestive than precise, the speed of propagation of disturbances is infinite. In this paper we develop a general theory of heat conduction for nonlinear materials with memory, a theory which has associated with it finite propagation speeds. In Section 3 we determine the restrictions that thermodynamics places on our constitutive relations. We show that our theory differs f rom other theories of heat conduction in that the heat-flux, like the entropy, is determined by the functional for the free-energy. In Section 6 we study the propagation of certain types of weak discontinuities. We show that in certain circumstances waves travelling in the direction of the heat-flux vector propagate faster than waves travelling in the opposite direction. In Section 7 we deduce the linearized theory appropriate to infinitesimal temperature gradients. We show that the linearized constitutive equation for the heat-flux q has the form: 1

We study the general nonlinear diffusion equation $${u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)}$$ that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters $${m > 1}$$ and $${0 < s < 1}$$ , we assume that the solutions are non-negative and that the problem is posed in the whole space. In this paper we prove the existence of weak solutions for all integrable initial data $${u_0 \ge 0}$$ and for all exponents $${m > 1}$$ by developing a new approximation method that allows one to treat the range $${m\geqq 3}$$ , which could not be covered by previous works. We also extend the class of initial data to include any non-negative measure $${\mu}$$ with finite mass. In passing from bounded initial data to measure data we make strong use of an L1- $${L^\infty}$$ smoothing effect and other functional estimates. Finite speed of propagation is established for all $${m \geqq 2}$$ , and this property implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for $${m < 2}$$ .

Vacuum solution and quasineutral limit of semiconductor drift-diffusion equation

It is proved that the solutions to the slow diffusion stochastic porous media equation $dX-{\Delta}( |X|^{m-1}X )dt=\sigma(X)dW_t,$ $ 1< m\le 5,$ in $\mathcal{O}\subset\mathbb{R}^d,\ d=1,2,3,$ have the property of finite speed of propagation of disturbances for $\mathbb{P}\text{-a.s.}$ ${\omega}\in{\Omega}$ on a sufficiently small time interval $(0,t({\omega}))$.

The aim of this paper is to review the main recent results about the dynamics of nonlinear partial differential equations describing flux-saturated transport mechanisms, eventually in combination with porous media flow and/or reactions terms. The result is a system characterized by the presence of wave fronts which move defining an interface. This can be used to model different process in applications in a variety of areas as Developmental Biology or Astrophysics. The concept of solution and its properties (well-posedness in a Bounded Variation scenario, Rankine–Hugoniot and geometric conditions for jumps, regularity results, finite speed of propagation, …), qualitative study of these fronts (traveling waves in particular) and application in morphogenesis cover the panorama of this review.

We focus on the thin film equation with lower order “backwards” diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a “porous media cutoff” of van der Waals forces. We treat in detail the equation u_t + \{u^n(u_{xxx}+ν u^{m–n}u_x–Au^{M–n}u_x)\}_x = 0, where ν = \pm 1 , n &gt; 0 , M &gt; m , and A ≥ 0 . Global existence of weak nonnegative solutions is proven when m–n &gt; –2 and A &gt; 0 or ν = –1 , and when –2 &lt; m–n &lt; 2 , A = 0 , ν = 1 . From the weak solutions, we get strong entropy solutions under the additional constraint that m–n &gt; –3/2 if ν = 1 . A local energy estimate is obtained when 2 ≤ n &lt; 3 under some additional restrictions. Finite speed of propagation is proven when m &gt; n/2 , for the case of “strong slippage”, 0 &lt; n &lt; 2 , when ν = 1 based on local entropy estimates, and for the case of “weak slippage”, 2 ≤ n &lt; 3 , when ν = \pm 1 based on local entropy and energy estimates.