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Register a new userExistence and extinction in finite time for Stratonovich gradient noise porous media equations
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Mattia Turra
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We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX - ( u Delta X + Delta psi (X) ) dt = sum_{i=1}^N langle b_i, abla X rangle circ dbeta_i$ in $]0,T[ times mathcal{O}$, with $X(0) = x(xi)$ in $mathcal{O}$ and $X = 0$ on $]0,T[ times partial mathcal{O}$. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.
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We establish finite time extinction with probability one for weak solutions of the Cauchy-Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate $O(t^frac{1}{2})$ whereas the support of solutions to the deterministic PME grows only with rate $O(t^{frac{1}{m{+}1}})$. The rigorous proof relies on a contraction principle up to time-dependent shift for Wong-Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.
The purpose of this paper is extend recent results of Bonder-Groisman and Foondun-Nualart to the stochastic wave equation. In particular, a suitable integrability condition for non-existence of global solutions is derived.
In this paper we investigate existence of solutions for the system: begin{equation*} left{ begin{array}{l} D^{alpha}_tu=textrm{div}(u abla p), D^{alpha}_tp=-(-Delta)^{s}p+u^{2}, end{array} right. end{equation*} in $mathbb{T}^3$ for $0< s leq 1$, and $0< alpha le 1$. The term $D^alpha_t u$ denotes the Caputo derivative, which models memory effects in time. The fractional Laplacian $(-Delta)^{s}$ represents the L{e}vy diffusion. We prove global existence of nonnegative weak solutions that satisfy a variational inequality. The proof uses several approximations steps, including an implicit Euler time discretization. We show that the proposed discrete Caputo derivative satisfies several important properties, including positivity preserving, convexity and rigorous convergence towards the continuous Caputo derivative. Most importantly, we give a strong compactness criteria for piecewise constant functions, in the spirit of Aubin-Lions theorem, based on bounds of the discrete Caputo derivative.
We delve deeper into the study of semimartingale attractors that we recently introduced in Allouba and Langa cite{AL0}. In this article we focus on second order SPDEs of the Allen-Cahn type. After proving existence, uniqueness, and detailed regularity results for our SPDEs and a corresponding random PDE of Allen-Cahn type, we prove the existence of semimartingale global attractors for these equations. We also give some results on the finite dimensional asymptotic behavior of the solutions. In particular, we show the finite fractal dimension of this random attractor and give a result on determining modes, both in the forward and the pullback sense.
In order to analyze numerically inverse problems several techniques based on linear and nonlinear stability analysis are presented. These techniques are illustrated on the problem of estimating mobilities and capillary pressure in one-dimensional two-phase displacements in porous media that are performed in laboratories. This is an example of the problem of estimating nonlinear coefficients in a system of nonlinear partial differential equations.
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