Sharp Regularity for Weak Solutions to the Porous Medium Equation (original) (raw)

Local bounds of the gradient of weak solutions to the porous medium equation

Partial Differential Equations And Applications, 2023

Let u be a nonnegative, local, weak solution to the porous medium equation for m ≥ 2 in a space-time cylinder ΩT. Fix a point (xo, to) ∈ ΩT : if the average a def = Br (xo) u(x, to) dx > 0, then the quantity |∇u m−1 | is locally bounded in a proper cylinder, whose center lies at time to + a 1−m r 2. This implies that in the same cylinder the solution u is Hölder continuous with exponent α = 1 m−1 , which is known to be optimal. Moreover, u presents a sort of instantaneous regularisation, which we quantify.

Boundary Regularity for the Porous Medium Equation

Archive for Rational Mechanics and Analysis

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 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/superparabolic functions and weak sub/supersolutions.

Energy estimates and convergence of weak solutions of the porous medium equation

2021

We study the convergence of the weak solution of the porous medium equation with a type of Robin boundary conditions, by tuning a parameter either to zero or to infinity. The convergence is in the strong sense, with respect to the L^2-norm, and the limiting function solves the same equation with Neumann (resp. Dirichlet) boundary conditions when the parameter is taken to zero (resp. infinity). Our approach is to consider an underlying microscopic dynamics whose space-time evolution of the density is ruled by the solution of those equations and from this, we derive sufficiently strong energy estimates which are the keystone to the proof of our convergence result.

Sharp boundedness and continuity results for the singular porous medium equation

Israel Journal of Mathematics

We consider non-homogeneous, singular (0 < m < 1) parabolic equations of porous medium type of the form ut − div A(x, t, u, Du) = µ in E T , where E T is a space time cylinder, and µ is a Radon-measure having finite total mass µ(E T). In the range (N −2) + N < m < 1 we establish sufficient conditions for the boundedness and the continuity of u in terms of a natural Riesz potential of the right-hand side measure µ.

Regularity of solutions and interfaces of a generalized porous medium equation inR N

Annali di Matematica Pura ed Applicata, 1991

We consider the Cauchy problem /or the generalized porous medium equation ut = A~(u) where u = u(x, t), x e R ~ and t > O, and the initial datum u(x, O) is assumed to be nouuegative, integrable and to have compact support. The nonlinearity q~(u) is a C ~ ]unction de]ined /or u >1 0 which grows like a power o] u. Our assumptions generalize the porous medium case, ~(u) = u ~, m > 1, and also include the equation o] the Marshak waves. This problem has /inite speed o] propagation. We estimate the rate o/growth o/the support o/the solution with precise estimates /or t-~ 0 and t-> ~. Our main result deals with the regularity o/ the solutions. We show that after a certaiq~ time t o the pressure, de]ined by v = ~(u), with ~'(u) = q~(u)/u and ~(0) ~-O, is a Lipschitz-continuous ]unction o/ x and t and the interlace is a Lipschitz-continuous sur]ace in Rzv+~; the solution u is HSlder contin.

Regularity of the free boundary for the porous medium equation

Journal of the American Mathematical Society, 1998

We study the regularity of the free boundary for solutions of the porous medium equation u t = Δ u m u_{t}=\Delta u^{m} , m > 1 m >1 , on R 2 × [ 0 , T ] {\mathcal {R}}^{2} \times [0,T] , with initial data u 0 = u ( x , 0 ) u^{0}=u(x,0) nonnegative and compactly supported. We show that, under certain assumptions on the initial data u 0 u^{0} , the pressure f = m u m − 1 f=m\, u^{m-1} will be smooth up to the interface Γ = ∂ { u > 0 } \Gamma = \partial \{ u >0 \} , when 0 > t ≤ T 0>t\leq T , for some T > 0 T >0 . As a consequence, the free-boundary Γ \Gamma is smooth.

Lower semicontinuity of weak supersolutions to the porous medium equation

Proceedings of the American Mathematical Society, 2015

Weak supersolutions to the porous medium equation are defined by means of smooth test functions under an integral sign. We show that nonnegative weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. This shows that weak supersolutions belong to a class of supersolutions defined by a comparison principle.