Related papers: $L^p$-parabolic regularity and non-degenerate Orns…

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x) \partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where $(a_{ij})$ is…

We study the parabolic equation \begin{align} \notag &u_t(t,x)=a^{ij}(t)u_{x^ix^j}(t,x)+f(t,x), \quad (t,x) \in [0,T] \times \mathbf{R}^d \\ &u(0,x)=u_0(x) \label{main eqn} \end{align} with the full degeneracy of the leading coefficients,…

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind \[ \mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}\partial_{x_{i}x_{j}}^{2} +\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}% \] where $(a_{ij})…

We consider linear elliptic and parabolic equations with measurable coefficients and prove two types of $L_{p}$-estimates for their solutions, which were recently used in the theory of fully nonlinear elliptic and parabolic second order…

We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ \partial_t^\alpha u - Lu + \lambda u= f \quad \mathrm{in} \quad (0,T) \times \mathbb{R}^d,$$ where $\partial_t^\alpha u$ is the Caputo…

In this paper, we present an $L_q(L_p)$-regularity theory for parabolic equations of the form: $$ \partial_t u(t,x)=\mathcal{L}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0. $$ Here, $\mathcal{L}^{\vec{a},\vec{b}}(t)$ represents…

We establish the maximal regularity for nonautonomous Ornstein-Uhlenbeck operators in $L^p$-spaces with respect to a family of invariant measures, where $p\in (1,+\infty)$. This result follows from the maximal $L^p$-regularity for a class…

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space $L^p({\mathbb R}^d;{\mathbb R}^m)$, $(d,m \ge 1)$ with $p\in [1,+\infty)$. Sufficient conditions for the associated evolution operator ${\bf…

In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators $$ \partial_{t}u(t,x) = \mathcal{L}^{a}u(t,x) + f(t,x), \quad t>0 $$ in $L_{q}(L_{p})$ spaces. Our spatial…

Let $z = (x,y) \in {\mathbb R}^d \times {\mathbb R}^{N-d}$, with $1 \le d < N$. We prove a priori estimates of the following type :$$\|\Delta\_{x}^{\frac \alpha 2} v \|\_{L^p({\mathbb R}^N)} \lec\_p\Big \| L\_{x } v +…

We consider a second-order parabolic equation in $\bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally H\"older continuous in the space variables.…

Suppose that ${\cal L}$ is a divergence form differential operator of the form ${\cal L}f:=(1/2) e^{U}\nabla_x\cdot\big[e^{-U}(I+H)\nabla_x f\big]$, where $U$ is scalar valued, $I$ identity matrix and $H$ an anti-symmetric matrix valued…

We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau…

We consider degenerate Kolmogorov-Fokker-Planck operators $$ \mathcal{L}u=\sum_{i,j=1}^{q}a_{ij}(x,t)\partial_{x_{i}x_{j}}^{2}u+\sum_{k,j=1}^{N}b_{jk}x_{k}\partial_{x_{j}}u-\partial_{t}u,\qquad (x,t)\in\mathbb{R}^{N+1},N\geq q\geq1 $$ such…

In this article we prove the BMO-$L_{\infty}$ estimate $$ \|(-\Delta)^{\gamma/2} u\|_{BMO(\mathbf{R}^{d+1})}\leq N \|\frac{\partial}{\partial t}u-A(t)u\|_{L_{\infty}(\mathbf{R}^{d+1})}, \quad \forall\, u\in C^{\infty}_c(\mathbf{R}^{d+1}) $$…

In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert…

This paper investigates weighted mixed-norm estimates for divergence-type parabolic equations on Reifenberg-flat domains with the conormal derivative boundary condition. The leading coefficients are assumed to be merely measurable in the…

In this work, we obtain quantitative estimates of the continuity constant for the $L^p$ maximal regularity of relatively continuous nonautonomous operators $\mathbb{A} : I \longrightarrow \mathcal{L}(D,X)$, where $D \subset X$ densely and…

The aim of this paper is to establish new pointwise regularity results for solutions to degenerate second order partial differential equations with a Kolmogorov-type operator of the form $$\mathscr{L} :=\sum_{i,j=1}^m \partial^2_{x_i x_j }…

We establish sharp interior and boundary regularity estimates for solutions to $\partial_t u - L u = f(t, x)$ in $I\times \Omega$, with $I \subset \mathbb{R}$ and $\Omega \subset\mathbb{R}^n$. The operators $L$ we consider are…