Probabilistic approach for systems of second order quasi-linear parabolic PDEs
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.
scientific article; zbMATH DE number 6009588