We prove that local weak solutions of the orthotropic $p$-harmonic equation in ${ℝ}^2$ are $C^1$ functions.

The goal of this article is to draw new applications of small-scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius $r\left(\lambda \right)\to 0$ then one can achieve improvements on the recent upper bounds of Logunov (2016) and Logunov and Malinnikova (2016) on the size of nodal sets, according to a certain power of $r\left(\lambda \right)$. We also show that the doubling estimates and the order-of-vanishing results of Donnelly and Fefferman (1988, 1990) can be improved. Due to results of Han (2015) and Hezari and Rivière (2016), small-scale QE holds on negatively curved manifolds at logarithmically shrinking rates, and thus we get logarithmic improvements on such manifolds for the above measurements of eigenfunctions. We also get $o\left(1\right)$ improvements for manifolds with ergodic geodesic flows. Our results work for a full density subsequence of any given orthonormal basis of eigenfunctions.

For all rank-2 Toda systems with an arbitrary singular source, we use a unified approach to prove:

1. The pair of local masses $\left({\sigma }_1,{\sigma }_2\right)$ at each blowup point has the expression $${\sigma }_i=2\left(N_{i1}{\mu }_1+N_{i2}{\mu }_2+N_{i3}\right),$$ where $N_{ij}\in \mathbb{Z}$, $i=1,2$, $j=1,2,3$.

2. At each vortex point $p_t$ if $\left({\alpha }_t^1,{\alpha }_t^2\right)$ are integers and ${\rho }_i\notin 4\pi \mathbb{N}$, then all the solutions of Toda systems are uniformly bounded.

3. If the blowup point $q$ is a vortex point $p_t$ and ${\alpha }_t^1,{\alpha }_t^2$ and $1$ are linearly independent over $Q$, then $$u^k\left(x\right)+2log|x-p_t|\le C.$$

The Harnack-type inequalities of 3 are important for studying the bubbling behavior near each blowup point.

The question of whether the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only magnetic diffusion can develop a finite-time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. In this paper, we derive a regularity criterion less restrictive than the Beale–Kato–Majda (BKM) regularity criterion type, namely any solution $\left(u,b\right)\in C\left(\left[0,T\right[;H^r\left({ℝ}^2\right)\right)$ with $r>2$ remains in $H^r\left({ℝ}^2\right)$ up to time $T$ under the assumption that

$${\int }_0^T\frac{\parallel \nabla u\left(t\right){\parallel }_{\infty }^{\frac{1}{2}}}{log\left(e+\parallel \nabla u\left(t\right){\parallel }_{\infty }\right)}dt<+\infty .$$

This regularity criterion may stand as a great improvement over the usual BKM regularity criterion, which states that if ${\int }_0^T\parallel \nabla \times u\left(t\right){\parallel }_{\infty }dt<+\infty$ then the solution $\left(u,b\right)\in C\left(\left[0,T\right[;H^r\left({ℝ}^2\right)\right)$ with $r>2$ remains in $H^r\left({ℝ}^2\right)$ up to time $T$. Furthermore, our result applies also to a class of equations arising in hydrodynamics and studied by Elgindi and Masmoudi (2014) for their $L^{\infty }$ ill-posedness.

We prove a bilinear Strichartz-type estimate for irrational tori via a decoupling-type argument, as used by Bourgain and Demeter (2015), recovering and generalizing a result of De Silva, Pavlović, Staffilani and Tzirakis (2007). As a corollary, we derive a global well-posedness result for the cubic defocusing NLS on two-dimensional irrational tori with data of infinite energy.

We provide a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form ${\partial }_{t}u+\mathcal{ℒ}{u}^m=0$, $m>1$, where the operator $\mathcal{ℒ}$ belongs to a general class of linear operators, and the equation is posed in a bounded domain $\Omega \subset {ℝ}^N$. As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, $\mathcal{ℒ}$ can be a fractional power of a uniformly elliptic operator with $C^1$ coefficients. Since the nonlinearity is given by $u^m$ with $m>1$, the equation is degenerate parabolic.

The basic well-posedness theory for this class of equations was recently developed by Bonforte and Vázquez (2015, 2016). Here we address the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our results cover also the local case when $\mathcal{ℒ}$ is a uniformly elliptic operator, and provide new estimates even in this setting.

A surprising aspect discovered in this paper is the possible presence of nonmatching powers for the long-time boundary behavior. More precisely, when $\mathcal{ℒ}={\left(-\Delta \right)}^s$ is a spectral power of the Dirichlet Laplacian inside a smooth domain, we can prove that

• when $2s>1-1∕m$, for large times all solutions behave as ${dist}^{1∕m}$ near the boundary;

• when $2s\le 1-1∕m$, different solutions may exhibit different boundary behavior.

This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the semilinear elliptic equation $\mathcal{ℒ}{u}^m=u$.

We consider a wave equation in three space dimensions, with a power-like nonlinearity which is either focusing or defocusing. The exponent is greater than 3 (conformally supercritical) and not equal to 5 (not energy-critical). We prove that for any radial solution which does not scatter to a linear solution, an adapted scale-invariant Sobolev norm goes to infinity at the maximal time of existence. The proof uses a conserved generalized energy for the radial linear wave equation, new Strichartz estimates adapted to this generalized energy, and a bound from below of the generalized energy of any nonzero solution outside wave cones. It relies heavily on the fact that the equation does not have any nontrivial stationary solution. Our work yields a qualitative improvement on previous results on energy-subcritical and energy-supercritical wave equations, with a unified proof.

We prove the existence of global $L^2$ weak solutions for a family of generalized inviscid surface quasigeostrophic (SQG) equations in bounded domains of ${ℝ}^2$. In these equations, the active scalar is transported by a velocity field which is determined by the scalar through a more singular nonlocal operator compared to the SQG equation. The result is obtained by establishing appropriate commutator representations for the weak formulation together with good bounds for them in bounded domains.

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector ${\chi }_{\left(-\infty ,E\right]}\left(H_L\right)$ of a Schrödinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded potential. Previously, such estimates were known only for individual eigenfunctions and for spectral projectors ${\chi }_{\left(E-\gamma ,E\right]}\left(H_L\right)$ with small $\gamma$. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. Our main application of such an estimate is to find lower bounds for the lifting of eigenvalues under semidefinite positive perturbations, which in turn can be applied to derive a Wegner estimate for random Schrödinger operators with nonlinear parameter-dependence. Another application is an estimate of the control cost for the heat equation in a multiscale domain in terms of geometric model parameters. Let us emphasize that previous uncertainty principles for individual eigenfunctions or spectral projectors onto small intervals were not sufficient to study such applications.