We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ partial_t^alpha u - Lu= f quad mathrm{in} quad (0,T) times mathbb{R}^d,$$ where $partial_t^alpha u$ is the Caputo fractional derivative of order $alph
a in (0,1]$, $Tin (0,infty)$, and $$Lu(t,x) := int_{ mathbb{R}^d} bigg( u(t,x+y)-u(t,x) - ycdot abla_xu(t,x)chi^{(sigma)}(y)bigg)K(t,x,y),dy $$ is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel $K$ with respect to $t$ and $y$. We also derive a weighted mixed-norm estimate for the equations with operators that are local in time, i.e., $alpha = 1$, which extend the previous results by using a quite different method.
In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $mathbb{R}^d$ and a complete Riemannian manifold $mathrm{M}$. On one hand, in $mathbb{R}^d$, we prove that any solution $u=u(t,x)$ t
o $u_t(t,x)-mathrm{L}_alpha^{kappa} u(t,x)=0$, where $mathrm{L}_alpha^{kappa}$ is a nonlocal operator of order $alpha$, is time analytic in $(0,1]$ if $u$ satisfies the growth condition $|u(t,x)|leq C(1+|x|)^{alpha-epsilon}$ for any $(t,x)in (0,1]times mathbb{R}^d$ and $epsilonin(0,alpha)$. We also obtain pointwise estimates for $partial_t^kp_alpha(t,x;y)$, where $p_alpha(t,x;y)$ is the fractional heat kernel. Furthermore, under the same growth condition, we show that the mild solution is the unique solution. On the other hand, in a manifold $mathrm{M}$, we also prove the time analyticity of the mild solution under the same growth condition and the time analyticity of the fractional heat kernel, when $mathrm{M}$ satisfies the Poincare inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in $mathbb{R}^d$ using the method of Fourier transform and contour integrals. We find that when $alphain (0,1]$, the fractional heat kernel is time analytic at $t=0$ when $x eq 0$, which differs from the standard heat kernel. As corollaries, we obtain sharp solvability condition for the backward fractional heat equation and time analyticity of some nonlinear fractional heat equations with power nonlinearity of order $p$. These results are related to those in [8] and [11] which deal with local equations.
We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection-diffusion equation [ partial_t theta - Delta theta + b cdot abla theta = 0 ] to satisfy local boundedness, a single-scale Harnack inequality,
and upper bounds on fundamental solutions. We demonstrate these properties for drifts $b$ belonging to $L^q_t L^p_x$, where $frac{2}{q} + frac{n}{p} < 2$, or $L^p_x L^q_t$, where $frac{3}{q} + frac{n-1}{p} < 2$. For steady drifts, the condition reduces to $b in L^{frac{n-1}{2}+}$. The space $L^1_t L^infty_x$ of drifts with `bounded total speed is a borderline case and plays a special role in the theory. To demonstrate sharpness, we construct counterexamples whose goal is to transport anomalous singularities into the domain `before they can be dissipated.
We study the degenerate Kolmogorov equations (also known as kinetic Fokker-Planck equations) in nondivergence form. The leading coefficients $a^{ij}$ are merely measurable in $t$ and satisfy the vanishing mean oscillation (VMO) condition in $x, v$ wi
th respect to some quasi-metric. We also assume boundedness and uniform nondegeneracy of $a^{ij}$ with respect to $v$. We prove global a priori estimates in weighted mixed-norm Lebesgue spaces and solvability results. We also show an application of the main result to the Landau equation. Our proof does not rely on any kernel estimates.
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 $mathbb R^d_+ = {x in mathb
b R^d,:, x_d>0}$ 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 stationary Stokes systems in divergence form with piecewise Dini mean oscillation coefficients and data in a bounded domain containing a finite number of subdomains with $C^{1,rm{Dini}}$ boundaries. We prove that if $(u, p)$ is a weak soluti
on of the system, then $(Du, p)$ is bounded and piecewise continuous. The corresponding results for stationary Navier-Stokes systems are also established, from which the Lipschitz regularity of the stationary $H^1$-weak solution in dimensions $d=2,3,4$ is obtained.
We study a class of elliptic and parabolic equations in non-divergence form with singular coefficients in an upper half space with the homogeneous Dirichlet boundary condition. Intrinsic weighted Sobolev spaces are found in which the existence and un
iqueness of strong solutions are proved when the partial oscillations of coefficients in small parabolic cylinders are small. Our results are new even when the coefficients are constants
In this note, we obtain a version of Aleksandrovs maximum principle when the drift coefficients are in Morrey spaces, which contains $L_d$, and when the free term is in $L_p$ for some $p<d$.
We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obta
in the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.
The free boundary problem for a two-dimensional fluid filtered in porous media is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the
initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong $L^infty_t L^2_x$ sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.
