arXiv subjects

Petra Wittbold

Publications and source records attributed to Petra Wittbold.

3 recordsLinked to original sources

arXiv2023

Entropy solutions for time-fractional porous medium type equations

In this paper we prove existence of entropy solutions to the time-fractional porous medium type equation, $$\partial_t[k\ast(u-u_0)]-\operatorname{div} (A(t,x)\nablaφ(u))=f\text{ in }Q_T=(0,T)\timesΩ,$$ with Dirichlet boundary condition, initial condition $u(0,\cdot)=u_0$ in $Ω$, and $L^1$-data $f\in L^1((0,T)\timesΩ), u_0\in L^1(Ω)$. To this end we approximate the data by $L^\infty$-functions, use a known existence result of weak solutions for these more regular data, and additionally a known contraction principle for weak solutions, which can be adopted to the entropy solutions.

math.AP↗

arXiv2020

Bounded weak solutions of time-fractional porous medium type and more general nonlinear and degenerate evolutionary integro-differential equations

We prove existence of a bounded weak solution to a degenerate quasilinear subdiffusion problem with bounded measurable coefficients that may explicitly depend on time. The kernel in the involved integro-differential operator w.r.t. time belongs to the large class of ${\cal PC}$ kernels. In particular, the case of a fractional time derivative of order less than 1 is included. A key ingredient in the proof is a new compactness criterion of Aubin-Lions type which involves function spaces defined in terms of the integro-differential operator in time. Boundedness of the solution is obtained by the De Giorgi iteration technique. Sufficiently regular solutions are shown to be unique by means of an $L_1$-contraction estimate.

math.AP↗

arXiv2014

Multi-dimensional scalar balance laws with discontinuous flux

We consider the problem of existence of entropy weak solutions to scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect both to the space variable x and the unknown quantity u. The problem is formulated in the framework of multi-valued mappings. We use the notion of entropy-measure valued solutions to prove the so-called contraction principle and comparison principle.

math.AP↗