Ansatze are constructed under which the solutions of 11-dimensional supergravity must be stationary points of a parabolic flow on a Riemannian manifold $M^{10-p}$. This parabolic flow turns out to be the Ricci flow coupled to a scalar field, a $\left(3-p\right)$-form, and a $4$-form. This allows the introduction of techniques from parabolic partial differential equations to the search of solutions to 11-dimensional supergravity. As a first step, Shi-type estimates and criteria for the long-time existence of the flow are established.

Consider the stationary Boltzmann equation in 2-dimensional convex domains with diffusive boundary condition. We establish the hydrodynamic limits while the boundary layers are present, and derive the steady Navier–Stokes–Fourier system with nonslip boundary conditions. Our contribution focuses on novel weighted $W^{1,\infty }$ estimates for the $𝜖$-Milne problem with geometric correction. Also, we develop stronger remainder estimates based on an $L^{2m}$-$L^{\infty }$ framework.

Let $u$ denote a solution to a rotationally invariant Hessian equation $F\left(D^{2}u\right)=0$ on a bounded simply connected domain $\mathrm{\Omega }\subset {ℝ}^2$, with constant Dirichlet and Neumann data on $\partial \mathrm{\Omega }$. We prove that if $u$ is real analytic and not identically zero, then $u$ is radial and $\mathrm{\Omega }$ is a disk. The fully nonlinear operator $F\not\equiv 0$ is of general type and, in particular, not assumed to be elliptic. We also show that the result is sharp, in the sense that it is not true if $\mathrm{\Omega }$ is not simply connected, or if $u$ is $C^{\infty }$ but not real-analytic.

We introduce Calderón–Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a Hörmander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the $A_2$-conjecture. The results are applied to obtain $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on ${ℝ}^d$ and smooth and angular domains.

Krein–de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We further develop this area by giving a description of canonical Hamiltonian systems whose spectral measures have logarithmic integral converging over the real line. This result can be viewed as a spectral version of the classical Szegő theorem in the theory of polynomials orthogonal on the unit circle. It extends the Krein–Wiener completeness theorem, a key fact in the prediction of stationary Gaussian processes.

We consider the energy-critical heat equation in ${ℝ}^n$ for $n\ge 6$

$$\left\{\begin{array}{c}{u}_t=\mathrm{\Delta }u+|u{|}^{\frac{4}{n-2}}u\text{ in }{ℝ}^n\times \left(0,\infty \right),\hfill \\ u\left(\cdot ,0\right)=u_0\text{ in }{ℝ}^n,\hfill \end{array}\right.$$

which corresponds to the $L^2$-gradient flow of the Sobolev-critical energy

$$J(u)={\displaystyle {\int }_{{ℝ}^n}{}_{}e[u],e[u]:=\frac{1}{2}|\nabla u{|}^2-\frac{n-2}{2n}|u{|}^{\frac{2n}{n-2}}}.$$

Given any $k\ge 2$ we find an initial condition $u_0$ that leads to sign-changing solutions with multiple blow-up at a single point (tower of bubbles) as $t\to +\infty$. It has the form of a superposition with alternate signs of singularly scaled Aubin–Talenti solitons,

$$u\left(x,t\right)=\sum _{j=1}^k{\left(-1\right)}^{j-1}{\mu }_j^{-\frac{n-2}{2}}U\left(\frac{x}{{\mu }_j}\right)+o\left(1\right)\text{ as }t\to +\infty ,$$

where $U\left(y\right)$ is the standard soliton

$$U\left(y\right)={\alpha }_n{\left(\frac{1}{1+|y{|}^2}\right)}^{\frac{n-2}{2}}$$

and

$${\mu }_j\left(t\right)={\beta }_j{t}^{-{\alpha }_j},{\alpha }_j=\frac{1}{2}\left({\left(\frac{n-2}{n-6}\right)}^{j-1}-1\right)$$

if $n\ge 7$. For $n=6$, the rate of the ${\mu }_j\left(t\right)$ is different and it is also discussed. Letting ${\delta }_0$ be the Dirac mass, we have energy concentration of the form

$$e\left[u\left(\cdot ,t\right)\right]-e\left[U\right]\rightharpoonup \left(k-1\right)S_n{\delta }_0\text{ as }t\to +\infty ,$$

where $S_n=J\left(U\right)$. The initial condition can be chosen radial and compactly supported. We establish the codimension $k+n\left(k-1\right)$ stability of this phenomenon for perturbations of the initial condition that have space decay $u_0\left(x\right)=O\left(|x{|}^{-\alpha }\right)$, , which yields finite energy of the solution.

We consider the singular set in the thin obstacle problem with weight $|x_{n+1}{|}^a$ for $a\in \left(-1,1\right)$, which arises as the local extension of the obstacle problem for the fractional Laplacian (a nonlocal problem). We develop a refined expansion of the solution around its singular points by building on the ideas introduced by Figalli and Serra to study the fine properties of the singular set in the classical obstacle problem. As a result, under a superharmonicity condition on the obstacle, we prove that each stratum of the singular set is locally contained in a single $C^2$ manifold, up to a lower-dimensional subset, and the top stratum is locally contained in a $C^{1,\alpha }$ manifold for some if .

In studying the top stratum, we discover a dichotomy, until now unseen, in this problem (or, equivalently, the fractional obstacle problem). We find that second blow-ups at singular points in the top stratum are global, homogeneous solutions to a codimension-2 lower-dimensional obstacle problem (or fractional thin obstacle problem) when , whereas second blow-ups at singular points in the top stratum are global, homogeneous, and $a$-harmonic polynomials when $a\ge 0$. To do so, we establish regularity results for this codimension-2 problem, which we call the very thin obstacle problem.

Our methods extend to the majority of the singular set even when no sign assumption on the Laplacian of the obstacle is made. In this general case, we are able to prove that the singular set can be covered by countably many $C^2$ manifolds, up to a lower-dimensional subset.