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Zeev Sobol

Publications and source records attributed to Zeev Sobol.

13 recordsLinked to original sources
arXiv2024

Stability Analysis of Degenerate Einstein Model of Brownian Motion

Our Recent advancements in stochastic processes have illuminated a paradox associated with the Einstein model of Brownian motion. The model predicts an infinite propagation speed, conflicting with the second law of thermodynamics. The modified model successfully resolves the issue, establishing a finite propagation speed by introducing a concentration-dependent diffusion matrix. In this paper, we outline the necessary conditions for this property through a counter-example. The second part of the paper focuses on the stability analysis of the solution of the degenerate Einstein model. We introduce a functional dependence on the solution that satisfies a specific ordinary differential inequality. Our investigation explores the solution's dependence on the boundary and initial data of the original problem, demonstrating asymptotic stability under various conditions. These results have practical applications in understanding stochastic processes within bounded domains.

math.AP↗
arXiv2022

Einstein model of the movement of small particles in a stationary liquid revisited: Finite Propagation Speed

The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix dependent of the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentration. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality and the Ladyzhenskaya-Uraltseva iteration lemma.

math.AP↗
arXiv2022

An Iterative Energy Estimate for Degenerate Einstein model of Brownian motion

We consider the degenerate Einsteins Brownian motion model when the time interval of the moving particles before the collisions, is reciprocal to the number of particles per unit volume u(x,t), at the point of observation x at time t. The parameter 0 < tau < C, which controls the characteristics of the fluid, almost increases unboundedly, as u approaches 0. This degeneration leads to the localization of the particle distribution in the media. In the paper, we present a structural condition of the time interval and the frequency of these free jumps, as functions of u which guarantees the finite speed of propagation of u.

math.AP↗
arXiv2022

An Iterative Energy Estimate for Degenerate Einstein model of Brownian motion

We consider the degenerate Einstein's Brownian motion model for the case when the time interval ($τ$) of particle Jumps before collision (free jumps) reciprocal to the number of particles per unit volume $u(x,t) > 0$ at the point of observation $x$ at time $t$. The parameter $0 < τ\leq C < \infty$, controls characteristic of the fluid "almost decreases" to $ 0 $ when $u \rightarrow \infty$. This degeneration leads to the localisation of the spread of particle propagation in the media. In our report we will present a structural condition of the time interval of free jumps - $τ$ and the frequency of these free jumps $ϕ$ as functions of $u$ which guarantees the finite speed of propagation of $u$.

math.AP↗
arXiv2017

On the $L^p$-theory of $C_0$-semigroups associated with second-order elliptic operators with complex singular coefficients

We study $L^p$-theory of second-order elliptic divergence type operators with complex measurable coefficients. The major aspect is that we allow complex coefficients in the main part of the operator, too. We investigate generation of analytic $C_0$-semigroups under very general conditions on the coefficients, related to the notion of form-boundedness. We determine an interval $J$ in the $L^p$-scale, not necessarily containing $p=2$, in which one obtains a consistent family of quasi-contractive semigroups. This interval is close to optimal, as shown by several examples. In the case of uniform ellipticity we construct a family of semigroups in an extended range of $L^p$-spaces, and we prove $p$-independence of the analyticity sector and of the spectrum of the generators.

math.AP↗
arXiv2012

Singular solutions for second-order non-divergence type elliptic inequalities in punctured balls

We study the existence and nonexistence of positive singular solutions to second-order non-divergence type elliptic inequalities with measurable coefficients. We prove the existence of a critical value $p^*$ that separates the existence region from non-existence. In the critical case $p=p^*$ we show that the existence of a singular solution depends on the rate at which the coefficients stabilize at zero and we provide some optimal conditions in this setting.

math.AP↗
arXiv2010

Gradient estimates for degenerate quasi-linear parabolic equations

For a general class of divergence type quasi-linear degenerate parabolic equations with differentiable structure and lower order coefficients form bounded with respect to the Laplacian we obtain $L^q$-estimates for the gradients of solutions, and for the lower order coefficients from a Kato-type class we show that the solutions are Lipschitz continuous with respect to the space variable.

math.AP↗
arXiv2010

Singular solutions to the heat equations with nonlinear absorption and Hardy potentials

We study the existence and nonexistence of singular solutions to the equation $u_t-Δu - \fracκ{|x|^2}u+|x|^αu|u|^{p-1}=0$, $p>1$, in $\R^N\times[0,\infty)$, $N\ge 3$, with a singularity at the point $(0,0)$, that is, nonnegative solutions satisfying $u(x,0)=0$ for $x\ne0$, assuming that $\a>-2$ and $κ<\left(\frac{N-2}2\right)^2$. The problem is transferred to the one for a weighted Laplace-Beltrami operator with a non-linear absorbtion, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a {\it source solution} with a multiple of the Dirac mass as initial datum, or as a unique {\it very singular solution}, leads to a complete classification of singular solutions to the original problem, which exist if and only if $p<1+\frac{2(2+α)}{N+2+\sqrt{(N-2)^2-4κ}}$.

math.AP↗
arXiv2006

Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations

We develop a new method to uniquely solve a large class of heat equations, so-called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of Stroock--Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier--Stokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.

math.PR↗
arXiv2005

Gradient Bounds for Solutions of Elliptic and Parabolic Equations

Let $L$ be a second order elliptic operator on $R^d$ with a constant diffusion matrix and a dissipative (in a weak sense) drift $b \in L^p_{loc}$ with some $p>d$. We assume that $L$ possesses a Lyapunov function, but no local boundedness of $b$ is assumed. It is known that then there exists a unique probability measure $μ$ satisfying the equation $L^*μ=0$ and that the closure of $L$ in $L^1(μ)$ generates a Markov semigroup $\{T_t\}_{t\ge 0}$ with the resolvent $\{G_λ\}_{λ> 0}$. We prove that, for any Lipschitzian function $f\in L^1(μ)$ and all $t,λ>0$, the functions $T_tf$ and $G_λf$ are Lipschitzian and |\nabla T_tf(x)| \leq T_t|\nabla f|(x) and |\nabla G_λf(x)| \leq \frac{1}λ G_λ|\nabla f|(x). An analogous result is proved in the parabolic case.

math.PR↗
arXiv2003

A critical phenomenon for sublinear elliptic equations in cone-like domains

We study positive supersolutions to an elliptic equation $(*)$: $-Δu=c|x|^{-s}u^p$, $p,s\in\bf R$ in cone-like domains in $\bf R^N$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_*<1$ such that equation $(*)$ has a positive supersolution if and only if $-\infty<p<p_*$. The value of $p_*$ is determined explicitly by $s$ and the geometry of the cone.

math.AP↗