Probabilistic approach for systems of second order quasi-linear parabolic PDEs
type: Item
instance of: scholarly article
title: Probabilistic approach for systems of second order quasi-linear parabolic PDEs
zbMATH Open document ID: 1246.35216
author: Jinxia Wang
author: Xicheng Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-02-27T00:00:00Z
review text: Based on a stochastic representation for second order quasilinear parabolic equations, the existence of local smooth solutions in Sobolev spaces is verified. The authors treat possibly degenerate or unbounded smooth coefficients. In particular the assumptions cover the viscous Burgers system in \(\mathbb R^d\). Using Bismut's formula, a global existence result for non-degenerate semilinear parabolic equations is derived. As an application, an upper bound for the rate of convergence for vanishing viscosity is studied. The general backward quasilinear parabolic system under consideration for \(u:\mathbb R^d \to \mathbb R^m\) is of the form (\(t\leq0\)) \[ \partial_t u +\frac12 \sum_{k,j,i} \sigma_k^i(u) \sigma_k^j(u) \partial_{i,j}^2 u + \sum_{i} b_i(u) \partial_i u =f(u)\;. \] The stochastic representation is \[ u(t,x)=\mathbb E u_0(X_{t,0}(x))- \mathbb E \int_t^0 f(u(r,X_{t,r}(x))dr, \] with the process \(X_{t,s}\) solving the following SDE: \[ X_{t,s}(x)= x + \int_t^s b(u(r,X_{t,r}(x))dr + \int_t^s \sigma(u(r,X_{t,r}(x))dW(r) \;. \] The functions \(b\) and \(\sigma\) might be unbounded, so only path-wise Sobolev regularity of \(X_{t,s}(x)\) is assumed.
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 6009588
zbMATH Keywords: stochastic representation
zbMATH Keywords: Bismut's formula
zbMATH Keywords: inviscid limit
zbMATH Keywords: global existence of solutions
zbMATH Keywords: local uniqueness of solutions
zbMATH Keywords: viscous Burgers system
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.09.059
OpenAlex ID: W1973914541
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DOI: 10.1016/J.JMAA.2011.09.059
description: scientific article; zbMATH DE number 6009588