Local boundedness of weak solutions to the Diffusive Wave Approximation of the Shallow Water equations

Singer, Thomas; Vestberg, Matias

doi:10.1016/j.jde.2018.08.051

Cited by 15 publications

(16 citation statements)

References 16 publications

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“…Using the same technique as in [30,Lemma 3.6], we conclude that any sub(super)-solution to (1.3) in the sense of Definition 1.1 satisfies the mollified version of (1.6),…”

Section: Mollification In Timementioning

confidence: 74%

Harnack’s inequality for doubly nonlinear equations of slow diffusion type

et al. 2021

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In this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

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“…Using the same technique as in [30,Lemma 3.6], we conclude that any sub(super)-solution to (1.3) in the sense of Definition 1.1 satisfies the mollified version of (1.6),…”

Section: Mollification In Timementioning

confidence: 74%

Harnack’s inequality for doubly nonlinear equations of slow diffusion type

et al. 2021

Self Cite

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“…In order to motivate the natural definition of weak solutions, we reformulate (2.2) 1 in terms of v := u − z. Formally applying the chain rule as in [24], we can write Eq. (2.2) 1 in the form…”

Section: Setting and Main Resultsmentioning

confidence: 99%

“…In the following, we explain the precise setting and present the natural definition of weak solutions which was introduced in [ 24 ]. For the right-hand side and the known function and the initial datum , we assume for some and where is given by ( 2.4 ).…”

Section: Setting and Main Resultsmentioning

confidence: 99%

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Existence of solutions to a diffusive shallow medium equation

2020

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“…In particular, evolutionary equations and systems can be used to model physical processes like heat conduction, diffusion processes or wave propagation, see e.g. [10,27,48]. The second interesting aspect here is the nonstandard growth setting.…”

Section: Introductionmentioning

confidence: 99%

Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

Arumugam

Erhardt

2020

J Elliptic Parabol Equ

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This paper deals with a homogeneous Neumann problem of a nonlinear diffusion system involving variable exponents dependent on spatial and time variables and cross-diffusion terms. We prove the existence of weak solutions using Galerkin’s approximation and we derive suitable energy estimates. To this end, we establish the needed Poincaré type inequality for variable exponents related to the Neumann boundary problem. Furthermore, we show that the investigated problem possesses a unique weak solution and satisfies a stability estimate, provided some additional assumptions are fulfilled. In addition, we show under which conditions the solution is nonnegative.

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Local boundedness of weak solutions to the Diffusive Wave Approximation of the Shallow Water equations

Cited by 15 publications

References 16 publications

Harnack’s inequality for doubly nonlinear equations of slow diffusion type

Harnack’s inequality for doubly nonlinear equations of slow diffusion type

Existence of solutions to a diffusive shallow medium equation

Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

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