On a singular incompressible porous media equation (original) (raw)

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.

Very weak solutions of singular porous medium equations with measure data

Communications on Pure and Applied Analysis, 2014

We consider non-homogeneous, singular (0 < m < 1) porous medium type equations with a non-negative Radon-measure µ having finite total mass µ(E T) on the right-hand side. We deal with a Cauchy-Dirichlet problem for these type of equations, with homogeneous boundary conditions on the parabolic boundary of the domain E T , and we establish the existence of a solution in the sense of distributions. Finally, we show that the constructed solution satisfies linear pointwise estimates via linear Riesz potentials.

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.

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.

A Steady Weak Solution of the Equations of Motion of a Viscous Incompressible Fluid through Porous Media in a Domain with a Non-Compact Boundary

2011

We assume that  (omega) is a domain in R2 or in R3 with a non-compact boundary, representing a generally inhomogeneous and anisotropic porous medium. We prove the weak solvability of the boundary-value problem, describing the steady motion of a viscous incompressible fluid in (omega) . We impose no restriction on sizes of the velocity fluxes through unbounded components of the boundary of (omega) . The proof is based on the construction of appropriate Galerkin approximations and study of their convergence. In Sect. 4, we provide several examples of concrete forms of (omega) and prescribed velocity profiles on ∂, when our main theorem can be applied.

Sharp Regularity for Weak Solutions to the Porous Medium Equation

arXiv: Analysis of PDEs, 2016

Let uuu be a nonnegative, local, weak solution to the porous medium equation for mge2m\ge2mge2 in a space-time cylinder OmegaT\Omega_TOmegaT. Fix a point (xo,to)inOmegaT(x_o,t_o)\in\Omega_T(xo,to)inOmegaT: 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 ∣nablaum−1∣|\nabla u^{m-1}|nablaum1 is locally bounded in a proper cylinder, whose center lies at time to+a1−mr2t_o+a^{1-m}r^2to+a1mr2. This implies that in the same cylinder the solution uuu is H\"older continuous with exponent alpha=frac1m−1\alpha=\frac1{m-1}alpha=frac1m1, which is known to be optimal. Moreover, uuu presents a sort of instantaneous regularisation, which we quantify.

A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions

Nonlinear Analysis, 2005

We study the localization properties of weak solutions to the Dirichlet problem for the degenerate parabolic equation (x,t) ∇u) = f, with variable exponent of nonlinearity . We prove the existence and uniqueness of weak solutions and establish conditions on the problem data and the exponent (x, t) sufficient for the existence of such properties as finite speed of propagation of disturbances, the waiting time effect, finite time vanishing of the solution. It is shown that the solution may instinct in a finite time even if ≡ (x) 0 in the problem domain but max = 0.

A priori estimates and regularization for a class of porous medium equations

The general class of porous medium equations: @S @t + r f(S)u ? r k(S)rS = Q(S) ; with di usion coe cient k vanishing for two values of saturation S and the fractional ow f having a characteristic`S'-shaped signature, is an accepted and physically reasonable model for describing two phase ow (formulated in terms of total velocity and pressure) of groundwater in certain common soils.