我们证明了尖锐的$L^p$估计的奇异传输方程的建设，我们称之为\emph{cascading solution}，该方程研究乘以有界函数和应用的希尔伯特变换的综合效果。在此过程中，我们证明了希尔伯特变换的不变性结果，这可能是独立的利益。最后，我们给出了一个有界的\emph{incompressible}速度场的例子，其中方程：$$\partial_t f +u\cdot\nabla f= H(f)$$发展急剧的$L^p$增长。我们研究的方程是相关的，作为模型，在流体方程的研究，以及在广义相对论。
英文摘要：
We prove sharp $L^p$ estimates for a singular transport equation by building what we call a \emph{cascading solution}; the equation studies the combined effect of multiplying by a bounded function and application of the Hilbert transform. Along the way we prove an invariance result for the Hilbert transform which could be of independent interest. Finally, we give an example of a bounded and \emph{incompressible} velocity field for which the equation:
$$\partial_t f +u\cdot\nabla f= H(f)$$ develops sharp $L^p$ growth. The equations we study are relevant, as models, in the study of fluid equations as well as in general relativity.
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some degenerate-singular Monge-Ampère type equations and prove the sharp boundary estimates for convex viscosity solutions of the problem. As a result, we obtain the optimal global Hölder regularity of the convex viscosity solutions.
This paper establishes sharp local regularity estimates for viscosity solutions of fully nonlinear parabolic free boundary problems with singular absorption terms. The main difficulties are due to the blow-up of the source along the free boundary and the lack of a variational structure. The proof combines the power of the Ishii-Lions method with intrinsically parabolic oscillation estimates. The results are new, even for second-order linear operators in nondivergence form.
In this paper we develop a boundary $\varepsilon$-regularity theory for optimal transport maps between bounded open sets with $C^{1,α}$-boundary. Our main result asserts sharp $C^{1,α}$-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions: The main quantitative assumptions are that the local nondimensionalized transport cost is small and that the boundaries are locally almost flat in $C^{1,α}$. Our method is completely variational and builds on the recently developed interior regularity theory.
We prove global gradient estimates for parabolic $p$-Laplace type equations with measure data, whose model is $$u_t - \textrm{div} \left(|Du|^{p-2} Du\right) = μ\quad \textrm{in} \ Ω\times (0,T) \subset \mathbb{R}^n \times \mathbb{R},$$ where $μ$ is a signed Radon measure with finite total mass. We consider the singular case $$\frac{2n}{n+1} <p \le 2-\frac{1}{n+1}$$ and give possibly minimal conditions on the nonlinearity and the boundary of $Ω$, which guarantee the regularity results for such measure data problems.
In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation $$ (-Δ)^σu = |x|^αu^p ~~~~~~~~~~~ in ~~ B_1 \backslash \{0\} $$ with an isolated singularity at the origin, where $σ\in (0, 1)$ and the punctured unit ball $B_1 \backslash \{0\} \subset \mathbb{R}^n$ with $n \geq 2$. When $-2σ< α< 2σ$ and $\frac{n+α}{n-2σ} < p < \frac{n+2σ}{n-2σ}$, we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of…
This paper is concerned with a class of multi-dimensional transport equations with nonlocal velocity. It is shown that the local smooth solution cannot exist globally in time via the De Giorgi iteration technique.
A theorem is proved on the uniform estimation of the residual term of the asymptotic expansion with respect to a small parameter of the solution of the initial problem for a singularly perturbed differential operator weakly nonlinear transport equation in the critical case.
By constructing appropriate smooth, possibly non-convex supersolutions, we establish sharp lower bounds near the boundary for the modulus of nontrivial solutions to singular and degenerate Monge-Ampère equations of the form $\det D^2 u =|u|^q$ with zero boundary condition on a bounded domain in $\mathbb{R}^n$. These bounds imply that currently known global Hölder regularity results for these equations are optimal for all $q$ negative, and almost optimal for $0\leq q\leq n-2$. Our study also establishes the optimality of global $C^{\frac{1}{n}}$ regularity for convex solutions to the Monge-Ampè…
A decomposition of a measurable function f by a one-sided local sharp maximal function and applications to one-sided operators
遵循Andrei Lerner在[ A pointwise estimate for the local sharp maximum function with applications to奇异积分“Bull.London Math.Soc.42（2010）843856]中的思想，我们获得了任意可测函数f在局部平均振荡方面的另一种分解。这使得我们可以得到新的估计涉及单边奇异积分和单边极大算子。作为这个结果的应用，我们得到了两个单侧奇异积分的加权不等式和一个关于可测函数f和sharp单侧算子的L（w）不等式.这些估计是更精确的意义上说，他们是有效的更大类的权重。
We obtain sharp local $C^{1,α}$ regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$ Δ_p u=γ(u-φ)^{γ-1}\,\text{ in }\,\{u>φ\}, $$ for $0<γ<1$ and $p\ge2$. At the free boundary $\partial\{u>φ\}$, we prove optimal $C^{1,τ}$ regularity of solutions, with $τ$ given explicitly in terms of $p$, $γ$ and smoothness of $φ$, which is new even in the linear setting.
In this paper, we consider a 1D periodic transport equation with nonlocal flux and fractional dissipation $$ u_{t}-(Hu)_{x}u_{x}+κΛ^αu=0,\quad (t,x)\in R^{+}\times S, $$ where $κ\geq0$, $0<α\leq1$ and $S=[-π,π]$. We first establish the local-in-time well-posedness for this transport equation in $H^{3}(S)$. In the case of $κ=0$, we deduce that the solution, starting from the smooth and odd initial data, will develop into singularity in finite time. If adding a weak dissipation term $κΛ^αu$, we also prove that the finite time blowup would occur.
In this paper, we revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity $L^1_t W^{1,p}$ for all $p<\infty$ in space dimensions $d\geq 2$ whose transport equations admit nonunique weak solutions belonging to $L^p_tC^k$ for all $p<\infty$ and $k\in \mathbb{N}$. In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna-Lions theory is sharp. The same result also holds for transport-diffusion equations with diffusion ope…
We consider the transport equation on $[0,T]\times \mathbb{R}^n$ in the situation where the vector field is $BV$ off a set $S\subset [0,T]\times \mathbb{R}^n$. We demonstrate that solutions exist and are unique provided that the set of singularities has a sufficiently small anisotropic fractal dimension and the normal component of the vector field is sufficiently integrable near the singularities. This result improves upon recent results of Ambrosio who requires the vector field to be of bounded variation everywhere. In addition, we demonstrate that under these conditions almost every trajecto…
The attracting inverse-square drift provides a prototypical counterexample to solvability of singular SDEs: if the coefficient of the drift is larger than a certain critical value, then no weak solution exists. We prove a positive result on solvability of singular SDEs where this critical value is attained from below (up to strict inequality) for the entire class of form-bounded drifts. This class contains e.g. the inverse-square drift, the critical Ladyzhenskaya-Prodi-Serrin class. The proof is based on a $L^p$ variant of De Giorgi's method.
We prove sharp partial regularity criteria of nonlinear potential theoretic nature for the Lebesgue-Serrin-Marcellini extension of nonhomogeneous singular multiple integrals featuring $(p,q)$-growth conditions.
We study the continuous version of a hyperbolic rescaling of a discrete game, called open mancala. The resulting PDE turns out to be a singular transport equation, with a forcing term taking values in $\{0,1\}$, and discontinuous in the solution itself. We prove existence and uniqueness of a certain formulation of the problem, based on a nonlocal equation satisfied by the free boundary dividing the region where the forcing is one (active region) and the region where there is no forcing (tail region). Several examples, most notably the Riemann problem, are provided, related to singularity forma…
