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search.filters.applied.f.access: openAccess × End date: 2022 × Start date: 2020 × Type: Text × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: phase transition ×

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Now showing 1 - 4 of 4
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    Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Lasarzik, Robert; Rocca, Elisabetta; Schimperna, Giulio
    In this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.
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    Analysis of a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Lasarzik, Robert
    In this paper, existence of generalized solutions to a thermodynamically consistent Navier--Stokes--Cahn--Hilliard model introduced in [19] is proven in any space dimension. The generalized solvability concepts are measure-valued and dissipative solutions. The measure-valued formulation incorporates an entropy inequality and an energy inequality instead of an energy balance in a nowadays standard way, the Gradient flow of the internal variable is fulfilled in a weak and the momentum balance in a measure-valued sense. In the dissipative formulation, the distributional relations of the momentum balance and the energy as well as entropy inequality are replaced by a relative energy inequality. Additionally, we prove the weak-strong uniqueness of the proposed solution concepts and that all generalized solutions with additional regularity are indeed strong solutions.
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    Continuum percolation in a nonstabilizing environment
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Jahnel, Benedikt; Jhawar, Sanjoy Kumar; Vu, Anh Duc
    We prove nontrivial phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson--Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox--Boolean model. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [MR2116736].
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    Bilinear coagulation equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Heydecker, Daniel; Patterson, Robert I.A.
    We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y) · Aπ (x) for a vector of conserved quantities π, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time tg ∈ (0,∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.