Local Hölder continuity of weak solutions to a diffusive shallow medium equation

An extensive study of the regularity of solutions to doubly singular equations

“…By contrast, in the slow diffusion case m + p > 3 which is not considered in this article, the inequality holds in the reverse direction, which means that no additional integrability is needed. For explicit calculations illustrating this point, consider the flat case of the equation studied in [18] and [19]. Earlier works treating the slow diffusion case (although not necessarily with the same definition) are [16] and [12].…”

Existence of solutions to a diffusive shallow medium equation

“…The article focused on the case , but the value of p does not really play a role in the arguments. In [ 25 ], local Hölder continuity was established for . Existence of solutions to a Cauchy–Dirichlet problem corresponding to ( 1.1 ) was proved in the case of trivial topography and slow diffusion in [ 3 ].…”

“…( 1.1 ) is used to model shallow water dynamics in situations such as floods and dam breaks; see [ 3 , 11 , 14 ]. Due to the variety of applications, we follow the terminology from [ 25 ] and use the term diffusive shallow medium equation or more concisely, DSM equation, to describe Eq. ( 1.1 ) for all .…”

Existence of Solutions to a Diffusive Shallow Medium Equation

“…The article focused on the case p < 2, but the value of p does not really play a role in the arguments. In [23] local Hölder continuity was established for p > 2. Existence of solutions to a Cauchy-Dirichlet problem corresponding to (1.1) was proved in the case of trivial topography z = 0 and slow diffusion α + p > 2 in [3].…”

“…In the range p < 2, equation (1.1) is used to model shallow water dynamics in situations such as floods and dam breaks, see [3,10,13]. Due to the variety of applications we follow the terminology from [23] and use the term diffusive shallow medium equation or more concisely, DSM equation, to describe equation (1.1) for all p > 1.…”