Local W -regularity Estimates for Weak Solutions of Parabolic Equations with Singular Divergence-free Drifts
نویسنده
چکیده
We study weighted Sobolev regularity of weak solutions of nonhomogeneous parabolic equations with singular divergence-free drifts. Assuming that the drifts satisfy some mild regularity conditions, we establish local weighted Lp-estimates for the gradients of weak solutions. Our results improve the classical one to the borderline case by replacing the L∞-assumption on solutions by solutions in the John-Nirenberg BMO space. The results are also generalized to parabolic equations in divergence form with small oscillation elliptic symmetric coefficients and therefore improve many known results.
منابع مشابه
The Harnack Inequality for Second-order Elliptic Equations with Divergence-free Drifts
in a domain Ω⊂R. Here a(x) is a given function and b(x) is a prescribed divergence free vector field, i.e., divb=0. The qualitative properties of solutions to elliptic and parabolic equations in divergence form with low regularity of the coefficients have been studied extensively, starting with the classical papers of De Giorgi [5], Nash [12], and Moser [11]. We are mostly interested in the imp...
متن کاملThe Harnack inequality for second-order parabolic equations with divergence-free drifts of low regularity
We establish the Harnack inequality for advection-diffusion equations with divergencefree drifts of low regularity. While our previous work [IKR] considered the elliptic case, here we treat the more challenging parabolic problem by adapting the classical Moser technique to parabolic equations with drifts with regularity lower than the scale-invariant spaces.
متن کاملOn Divergence-free Drifts
We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form ∂t−∆+ b ·∇ resp. −∆+ b ·∇ with a divergence-free drift b. We prove the Liouville theorem and Harnack inequality when b ∈ L∞(BMO) resp. b ∈ BMO−1 and provide a counterexample demonstra...
متن کاملA note on critical point and blow-up rates for singular and degenerate parabolic equations
In this paper, we consider singular and degenerate parabolic equations$$u_t =(x^alpha u_x)_x +u^m (x_0,t)v^{n} (x_0,t),quadv_t =(x^beta v_x)_x +u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)times (0,T)$, subject to nullDirichlet boundary conditions, where $m,n, p,qge 0$, $alpha, betain [0,2)$ and $x_0in (0,a)$. The optimal classification of non-simultaneous and simultaneous blow-up solutions is determin...
متن کاملLocal Regularity Results for Some Parabolic Equations
In this paper we prove the local L s regularity (where s depends on the summability of the data) for local " unbounded " weak solutions of a class of nonlinear parabolic equations including the p-Laplacian equation.
متن کامل