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Register a new userOn divergence form SPDEs with VMO coefficients in a half space
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Authors
N.V. Krylov
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We extend several known results on solvability in the Sobolev spaces $W^{1}_{p}$, $pin[2,infty)$, of SPDEs in divergence form in $bR^{d}_{+}$ to equations having coefficients which are discontinuous in the space variable.
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We present several results on solvability in Sobolev spaces $W^{1}_{p}$ of SPDEs in divergence form in the whole space.
This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous media equations, stochastic $p$-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.
We prove that the weak version of the SPDE problem begin{align*} dV_{t}(x) & = [-mu V_{t}(x) + frac{1}{2} (sigma_{M}^{2} + sigma_{I}^{2})V_{t}(x)]dt - sigma_{M} V_{t}(x)dW^{M}_{t}, quad x > 0, V_{t}(0) &= 0 end{align*} with a specified bounded initial density, $V_{0}$, and $W$ a standard Brownian motion, has a unique solution in the class of finite-measure valued processes. The solution has a smooth density process which has a probabilistic representation and shows degeneracy near the absorbing boundary. In the language of weighted Sobolev spaces, we describe the precise order of integrability of the density and its derivatives near the origin, and we relate this behaviour to a two-dimensional Brownian motion in a wedge whose angle is a function of the ratio $sigma_{M}/sigma_{I}$. Our results are sharp: we demonstrate that better regularity is unattainable.
We study a class of second-order degenerate linear parabolic equations in divergence form in $(-infty, T) times mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-infty, T) times partial mathbb R^d_+$, where $mathbb R^d_+ = {x in mathbb R^d,:, x_d>0}$ and $Tin {(-infty, infty]}$ is given. The coefficient matrices of the equations are the product of $mu(x_d)$ and bounded uniformly elliptic matrices, where $mu(x_d)$ behaves like $x_d^alpha$ for some given $alpha in (0,2)$, which are degenerate on the boundary ${x_d=0}$ of the domain. Under a partially VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.
We study a class of linear parabolic equations in divergence form with degenerate coefficients on the upper half space. Specifically, the equations are considered in $(-infty, T) times mathbb{R}^d_+$, where $mathbb{R}^d_+ = {x in mathbb{R}^d,:, x_d>0}$ and $Tin {(-infty, infty]}$ is given, and the diffusion matrices are the product of $x_d$ and bounded uniformly elliptic matrices, which are degenerate at ${x_d=0}$. As such, our class of equations resembles well the corresponding class of degenerate viscous Hamilton-Jacobi equations. We obtain wellposedness results and regularity type estimates in some appropriate weighted Sobolev spaces for the solutions.
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