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Register a new userProbabilistic representation for solutions of an irregular porous media type equation
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We consider a porous media type equation over all of $R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. One of the main analytic ingredients of the proof, is a new result on uniqueness of distributional solutions of a linear PDE on $R^1$ with non-continuous coefficients.
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The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized critical behaviour of stochastic nonlinear diffusion equations with critical states.
We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form $$mathrm{d} X= u(omega,t,X), mathrm{d} t + frac12 sigma(omega,t,X)sigma(omega,t,X),mathrm{d} t + sigma(omega,t,X) , mathrm{d}W(t), $$ where the drift coefficient $u$ is random and irregular. The random and regular noise coefficient $sigma$ may vanish. The main contribution is a pathwise uniqueness result under the assumptions that $u$ belongs to $L^p(Omega; L^infty([0,T];dot{H}^1(mathbb{R})))$ for any finite $pge 1$, $mathbb{E}left|u(t)-u(0)right|_{dot{H}^1(mathbb{R})}^2 to 0$ as $tdownarrow 0$, and $u$ satisfies the one-sided gradient bound $partial_x u(omega,t,x) le K(omega, t)$, where the process $K(omega,t )>0$ exhibits an exponential moment bound of the form $mathbb{E} expBig(pint_t^T K(s),mathrm{d} sBig) lesssim {t^{-2p}}$ for small times $t$, for some $pge1$. This study is motivated by ongoing work on the well-posedness of the stochastic Hunter--Saxton equation, a stochastic perturbation of a nonlinear transport equation that arises in the modelling of the director field of a nematic liquid crystal. In this context, the one-sided bound acts as a selection principle for dissipative weak solutions of the stochastic partial differential equation (SPDE).
We study an imbibition problem for porous media. When a wetted layer is above a dry medium, gravity leads to the propagation of the water downwards into the medium. In experiments, the occurrence of fingers was observed, a phenomenon that can be described with models that include hysteresis. In the present paper, we describe a single finger in a moving frame and set up a free boundary problem to describe the shape and the motion of one finger that propagates with a constant speed. We show the existence of solutions to the travelling wave problem and investigate the system numerically.
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.
We construct examples of solutions to the incompressible porous media (IPM) equation that must exhibit infinite in time growth of derivatives provided they remain smooth. As an application, this allows us to obtain nonlinear instability for a class of stratified steady states of IPM.
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