Papers

We study a new quasilinear system of balance laws (i.e. conservative first-order PDEs with source terms) for the numerical simulation by a front-capturing approach of viscoelastic athermal multi-material flows with sharp interfaces (like a mixture of immiscible fluids and solids). It naturally extends with an energy balance that is meaningful to stability and thermodynamics. In particular, the unequivocal definition of solutions to Cauchy problems can be guaranteed in some cases, thanks to a symmetric-hyperbolic formulation.

We study the loss of observability for a class of strongly coupled parabolic systems as the coupling parameter tends to zero. The system is observed through a fixed matrix $B$ on a measurable subset of space-time with positive measure. Under the Kalman rank condition, we determine the exact blow-up rates of the optimal observability cost. Different components of the state, as well as different combinations of them, may have different blow-up rates, and these rates are determined by the algebraic structure of the pair $(B,K)$. The full-state observability cost is determined by the largest of these rates. We also show that the worst-case minimum $L^\infty$-norm of null controls has the same asymptotic behavior.

This paper investigates the singular limit problem of the one-dimensional steady compressible Euler system with heat conduction in a finite nozzle. The existence of the heat conduction solutions were established and boundary layers at one or two nozzle endpoints occur as the coefficient of heat conductivity goes to zero. The asymptotic behavior of the solutions vary depending on the state at the entrance and the value of the pressure at the exit, which classify the solutions into purely supersonic solution, purely subsonic solution and transonic solution. Moreover, we perform a formal asymptotic expansion of solutions and derive the boundary layer equations.

We study the semilinear pseudo-relativistic equation \[ \sqrt{-\Delta+m^{2}}\,Q+\omega Q=Q^{p-1}\qquad\text{in }\R^{d}, \] with $m>0$, $\omega>0$, $d\ge1$ and $2<p<2^{*}:=\frac{2d}{d-1}$ (with $2^{*}=\infty$ when $d=1$). We prove that the positive radial variational ground state is unique. The proof rests on a mass-covariant oscillation theorem for the linearized operator, established through a purely variational reduction: the Caffarelli–Silvestre–Duffin extension for $\sqrt{-\Delta+m^{2}}$ combined with a ground-state substitution reduces the linearized eigenvalue problem to a massless, potential-free weighted Steklov problem, whose second eigenfunction changes sign exactly once by strict superadditivity of the weighted Dirichlet energy under sign decomposition. The nondegeneracy of the linearization is then obtained from two orthogonality relations; the failure of scale invariance for $m>0$, which destroys the second relation used in the massless Frank–Lenzmann–Silvestre theory, is repaired by a mass-covariance identity. Global uniqueness follows by a continuation argument in the mass parameter, anchored at $m=0$ by the theorem of Frank–Lenzmann–Silvestre.

We consider an initial value problem for a time-fractional evolution equation in a Hilbert space $X$: $$ \partial_t^{\alpha} (u(t)-a) = Au(t) \qquad \mbox{for $0<t<T$}, $$ where $\partial_t^{\alpha}$ denotes a fractional differential operator of Caputo type with order $0<\alpha<1,$ $u\colon (0,T) \to X,$ $a$ describes an initial value, and $A$, with domain $\mathcal{D}(A),$ is the generator of a contraction C$_0$ semigroup in $X$. We prove a fractional Hille-Yosida theorem characterizing the unique existence of a weak solution for every $a\in X$. We also discuss the unique existence of a strong solution for $a \in \mathcal{D}(A),$ as well as the continuity of weak solutions. Our results are applicable, for example, to time-fractional transport equations and Boltzmann equations. The proofs are mainly based on constructing an approximating sequence of solutions using the Yosida approximation.

We resolve an open problem posed by Frank–Loss. Let $n\ge 3$ and let $\varphi\in L^p(\mathbb{S}^n)$, with $\frac{n}{n-1}<p<\infty$, be a nontrivial Dirac zero mode on $\mathbb{S}^n$, i.e. a nonzero spinor satisfying \begin{equation*} D\varphi = iA\cdot\varphi, \end{equation*} where $D$ is the Dirac operator and $A$ is a vector field with $\mathrm{d} A^\flat\in L^{n/2}$. We prove the sharp lower bound \begin{equation*} \|\mathrm{d} A^\flat\|_{\frac{n}{2}} \ge 2\left[\frac{n}{2}\right]^{-\frac12}\frac{n-1}{n-2}S_n = \left[\frac{n}{2}\right]^{-\frac12}\frac{n(n-1)}{2}\omega_n^{\frac{2}{n}}. \end{equation*} Equality is attainable if and only if $n$ is odd; in that case, modulo conformal and gauge transformations, $\varphi$ is a Killing spinor and $A$ is a real multiple of the Reeb field associated with $\varphi$. The case $n=3$ was proved in our recent paper using a different method. In the paper we divide the remaining cases into 3 cases: i) $n\ge 5$ is odd, ii) $n\ge 5$ is even and iii) $n=4$. All these cases need to use different methods. For $n\ge 5$, the argument crucially reduces to an improved Sobolev inequality on $\mathbb{S}^n$ under a barycenter constraint. Specifically, for $u\in W^{1,2}(\mathbb{S}^n)$ we consider \begin{equation*} \mathfrak{a}_n := \inf\Bigg\{ \frac{ \int \Big(|\nabla u|^2 + \frac{n(n-2)}{4}u^2\Big) - \frac{n(n-2)}{4}\omega_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 }{ \omega_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 - \int u^2 } \,\Bigg|\, \int x|u|^{\frac{2n}{n-2}}=0,\ \omega_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 - \int u^2>0 \Bigg\}, \end{equation*} and we obtain the following universal estimate \begin{equation*} \mathfrak{a}_n > \frac{n(n-2)}{4(n^2-3n+1)}, \end{equation*} which is enough for our aim. Determining the exact value of $\mathfrak{a}_n$ remains open.

In this work, we provide a complete characterisation of Pitt's inequality for metaplectic operators. The symplectic geometry underlying a metaplectic operator distinguishes effective directions, along which its action is Fourier-type, from singular directions, along which concentration is preserved. For this reason, we adopt two complementary perspectives, thereby obtaining both an isotropic Pitt's inequality, emulating the classical theorem of Pitt, and an anisotropic inequality that adapts to the geometric features of the metaplectic group. As a by-product, we obtain Pitt's inequality for Fourier transforms along subspaces of $\rd$ and new quantitative time-dependent boundedness results for metaplectic operators on homogeneous Sobolev spaces, with applications to Schrödinger evolutions generated by quadratic Hamiltonians. Additionally, we derive logarithmic and entropic uncertainty principles for the metaplectic group. We show that the classical entropic uncertainty principle fails when both the effective and singular directions are present. In this case, a generally unbounded corrective entropy term accounts for the concentration-preserving directions and restores a meaningful lower bound. For quadratic Schrödinger evolutions, this correction follows the time-dependent geometry of the nondispersive directions and adjusts the uncertainty estimate accordingly.

We study bounded viscosity solutions of the degenerate or singular parabolic equation \[ u_t-|Du|^{\gamma}\Delta_p^{\mathrm{N}}u=f \qquad \text{in } Q_1, \] where \[ \Delta_p^{\mathrm{N}}u := \Delta u+(p-2) \left\langle D^2u\frac{Du}{|Du|}, \frac{Du}{|Du|} \right\rangle, \qquad -1<\gamma<\infty,\quad 1<p<\infty, \] and $f\in C^0(Q_1)\cap L^\infty(Q_1)$. Given any prescribed $\alpha\in(0,1)$, we prove that there exists $\varepsilon=\varepsilon(n,\alpha)>0$ such that \[ |p-2|+|\gamma|\leq\varepsilon \quad\Longrightarrow\quad u\in C_{\mathrm{loc}}^{1+\alpha,\frac{1+\alpha}{2}}(Q_1). \] In particular, the spatial gradient Hölder exponent can be chosen arbitrarily close to $ 1 $ as $(p,\gamma)\to(2,0)$. This extends the result of Andrade and Santos (Calc. Var. Partial Differential Equations 61, Paper No. 196, 2022) for $\gamma=0$ to the joint regime $(p,\gamma)\to(2,0)$ and answers affirmatively the question raised in Remark~1.1 therein.

On the upper hemisphere, we use the Obata-Escobar argument to establish an Obata theorem for $\sigma_k$-flat and positive constant boundary $\mathcal B_k$-curvature equation, which was initiated by Case-Wang [Adv. Math. 337 (2018), 83-106].

We consider the following singular Liouville equation $-\Delta v=\lambda V(x)|x|^2e^v \quad \text{in } B_1, \quad v=0 \quad \text{on } \partial B_1, $ where $B_1\subset\mathbb R^2$ is the unit disk, $\lambda>0$ is a small parameter, and $V$ is a positive smooth function. We first prove a non-degeneracy result for the bubbling solutions constructed in \cite{D-W-Z2026} by the local Pohozaev identities. Then we use this non-degeneracy result to establish the local uniqueness of bubbling solutions whose concentration parameters are sufficiently close.

In this short communication we answer negatively to a question posed by the authors in their work \cite[Section 1.4]{Brue2022} about the complex interpolation nature of the spaces $BV^{s}(\R^n)$, introduced by G.E. Comi and G. Stefani in \cite{ComiStefani2019}.

We solve the asymptotic Plateau problem in hyperbolic space for the curvature given by the normalized geometric mean of the $p$-fold sums of the principal curvatures. For every $n\ge3$, $2\le p\le n-1$, and $\sigma\in(0,1)$, each bounded domain $\Omega\subset\mathbb{R}^n$ with smooth mean-convex boundary admits a unique complete admissible vertical graph with prescribed curvature $\sigma$ and asymptotic boundary $\partial\Omega\times\{0\}$. The result extends the known cases $(n,p)=(3,2)$ and $(4,3)$ to the stated range. The key curvature estimate is uniform with respect to the positive boundary height in the approximating Dirichlet problems. The proof establishes two lower bounds for the full third-derivative quadratic form under the linearized constraints: a sharp universal bound with coefficient $2/p$ and a stronger direction-dependent bound when the relevant principal curvature lies in a specified range.

In this paper we completely classify the minimizers of a measure constraint global vectorial free boundary problem with a linear datum. More precisely, we show that there exists a unique minimizer whose coincidence set is an ellipsoid. We are motivated by the connection of the problem with the blow-ups of the vectorial Bernoulli free boundary problem and the obstacle problem.

For the fractional heat equation $\partial_t u + (-\Delta)^{\beta/2}u=0$ on $\mathbb{R}^d$ with $\beta\in(0,2)$, Weber and Zacher introduced the optimal constant $C_{\rm LY}(\beta,d)$ in the Li–Yau type estimate $$(-\Delta)^{\beta/2}\log u(t,\cdot)\leq \frac {C_{\rm LY}(\beta,d)}{t},\quad t>0.$$ They asked whether $C_{\rm LY}(\beta,d)$ converges to the classical Li–Yau constant $d/2$ as $\beta\uparrow2$. In this note we prove that the answer is affirmative. More precisely, for every fixed $d\geq1$ there exists a constant $K_d>0$ such that $$\frac{d}{\beta}\leq C_{\rm LY}(\beta,d)\leq\frac{d}{\beta}+K_d(2-\beta)\log\frac{e}{2-\beta},\quad 1\leq\beta<2.$$ Consequently, $\lim_{\beta\uparrow2}C_{\rm LY}(\beta,d)=d/2$. The proof has two main ingredients. First, the Bochner subordination formula, together with a concavity argument, shows that the supremum in Weber–Zacher's formula for $C_{\rm LY}(\beta,d)$ is attained at the origin. Second, the factor $2-\beta$ is derived from a lower bound for the fractional heat kernel profile in the large-distance regime, where the Lévy–Itô decomposition plays an important role.

We establish two critical one-component regularity criteria for the three-dimensional incompressible Navier–Stokes equations. First, for divergence-free initial data $u_0\in H^1(\mathbb R^3)$, we prove that a strong solution can be continued beyond a finite time $T$ provided that \[ u^3\in L^2\bigl(0,T;\dot B^{3/2}_{2,r}(\mathbb R^3)\bigr) \qquad\text{for some }2<r\leq\infty. \] Second, for every $2<p<\infty$, if $u_0\in\dot H^{1/2}(\mathbb R^3)$ and, for some fixed unit vector $\boldsymbol e\in\mathbb S^2$, \[ u\cdot\boldsymbol e \in L^p\bigl(0,T;\dot H^{1/2+2/p}(\mathbb R^3)\bigr), \] then the corresponding Fujita–Kato solution also extends beyond $T$, without any additional low-integrability assumption on the initial vorticity. These results extend the critical one-component regularity criteria of Han, Lei, Li, and Zhao in two different directions. At the time endpoint $p=2$, we combine a new vertical-vorticity estimate with coupled anisotropic energy estimates and an energy-dependent frequency decomposition to relax the dyadic square-summability condition \[ \dot H^{3/2}=\dot B^{3/2}_{2,2} \] to \[ \dot B^{3/2}_{2,r}, \qquad 2<r\leq\infty, \] with the endpoint case $r=\infty$ closed by an Osgood-type argument. In the nonendpoint range $2<p<\infty$, we use a positive-time heat-flow decomposition to separate the solution into a smooth linear part and a nonlinear remainder. The remainder has finite energy and acquires the required $L^q$-integrability of the vorticity at positive times, which allows the anisotropic energy estimates to be applied while the additional transport, stretching, and pressure terms generated by the heat flow remain integrable. This removes the extra $L^{q_0}$-integrability assumption on the initial vorticity while retaining the natural critical initial-data space $\dot H^{1/2}(\mathbb R^3)$.

We study the linearized dynamics of the Kolmogorov flow $U^{(0)}=(a\sin x_2,0)$ for the two-dimensional incompressible Navier–Stokes equations with horizontal dissipation $\nu\partial_1^2$. Unlike the fully dissipative case, this anisotropic system admits $U^{(0)}$ as an exact unforced steady state. On each horizontal Fourier mode $k$, the dissipation reduces to the scalar $-\nu k^2$ and commutes with the advection, so that the linearized semigroup factors exactly into $e^{-\nu k^2t}$ times the inviscid Euler group. For $|k|>1$, we prove two-sided bounds showing that the decay rate is exactly $\nu k^2$, uniformly in the shear amplitude $a$; hence no enhanced dissipation occurs. This reflects a fundamental mismatch: the shear transfers enstrophy to high vertical frequencies, which horizontal dissipation does not detect. At the critical modes $|k|=1$, the inviscid group grows like $\sqrt{2at}$, producing a transient amplification of size $(a/e\nu)^{1/2}$ before decay at the rate $\nu$. The horizontally independent modes form an infinite-dimensional undamped kernel. For $0<|k|<1$, the viscous spectrum is an exact translate of the inviscid one, and the mode is linearly unstable if and only if $a\Lambda(k)>\nu k^2$. In particular, when $L>2\pi$, the flow is linearly unstable for all sufficiently large $a$.

We prove mass estimates for PDE-constrained measures on sets of $\sigma$-finite ${\mathcal H}^m$-measure, assuming an appropriate rank condition. As an application, we derive a Michael–Simon inequality for varifolds with bounded anisotropic first variation with respect to an integrand that satisfies the natural atomic condition (AC1).

In this paper, we study bounded self-similar solutions for the exponential semilinear heat equation via the $F$-functional and entropy. Motivated by recent developments in mean curvature flow, we prove that every non-constant bounded solution of the self-similar equation is entropy unstable. We also show that the trivial profile $w\equiv0$ is quantitatively isolated among bounded solutions. These results provide both a variational instability theorem for nontrivial profiles and a rigidity theorem for the trivial profile.

In this paper, we consider a free boundary problem with fractional diffusion $(-\Delta)^s$ $(0<s<1)$, as a model for species spreading, which can be viewed as a natural extension of the free boundary model in \cite{CDLL2019}, where the nonlocal diffusion term is defined via a continuous integrable kernel function $J(x)$. This fractional diffusion model can also be viewed as a nonlocal version of the free boundary model in \cite{DuLin2010}, whose local diffusion term is given by the classical Laplacian. We first establish the global existence and uniqueness of the solution, which is the main contribution of this paper. This well-posedness question has been open for some time now due to the difficulties caused by the singularity and non-integrability of the associated kernel function of $(-\Delta)^s$, and the lack of general enough regularity results for operators extending the standard fractional Laplacian. A crucial step that enables us to answer this question relies on an approximation approach, which breaks an associated linear fractional parabolic initial boundary value problem with curved boundaries into a sequence of approximating problems, where the functions representing the curved boundaries are replaced by approximating step functions. We then show that, for Fisher-KPP type nonlinear growth terms, the long-time dynamics of the model exhibits a spreading-vanishing dichotomy, similar to the earlier models of \cite{DuLin2010, CDLL2019}. More precise descriptions of the spreading profile of the solution will be considered in a separate work.

The pseudo-parabolic term $u_{xxt}$ serves as a canonical example of higher-order viscosity that provides effective regularization while capturing the refined physical mechanism. How it alters the dynamics of prototype equations remains a fundamental issue, which is not yet fully understood. The goal of this paper is to show that this term induces a sharp transition in traveling wave structure, via studying the pseudo-parabolic Fisher-KPP equation $$ u_t - \tau u_{xxt} = D u_{xx} + u(1-u). $$ We completely characterize how the parameter ratio $\tau/D$ acts as a critical switch: it preserves the monotonic structure of classical traveling waves when $\tau/D \leq 1$, yet it triggers a qualitative shift to oscillatory traveling waves when $\tau/D > 1$. Numerical simulations support these findings. The threshold $\tau/D=1$ thus captures the dual role of the pseudo-parabolic term: it acts as a structure-preserving viscosity when small, but when large it reflects dominant capillary hysteresis, generating the oscillations that explain the saturation overshoot, a phenomenon that contradicts classical diffusion models yet is widely observed in two-phase flow.

We give an affirmative answer to the planar local smoothing problem in Question~1.7 of D.~Ntalampekos and positive and negative answers to the basic approximation and level-set parts of his higher-dimensional Question~1.8. In every dimension $n\ge2$, each continuous Lebesgue monotone function in $W^{1,p}$ on a bounded open set admits uniform and strong $W^{1,p}$ approximation by monotone $C^{1,\alpha}_{\loc}$ functions, with unchanged Sobolev boundary values and no increase of the $p$-Dirichlet energy, for $1<p<\infty$. The energy can be made strictly smaller unless the original function is $p$-harmonic. In the plane we obtain smooth local replacement at every isolated $p$-harmonic critical point, with arbitrarily small $C^1$ and Sobolev error. A point of gradient index $-m$ can be resolved into exactly $m$ nondegenerate saddles. Together with Ntalampekos's planar theorem, this gives smooth monotone density with fixed boundary values; at a nonsmooth $p$-harmonic minimizer the energy increase is unavoidable but can tend to zero. In dimensions $n\ge3$, an explicit Lipschitz monotone function has a nonmanifold point on every level in an interval, although it admits smooth monotone approximation. A product extension of a planar homogeneous $7$-harmonic function also rules out a general discrete exceptional set under the exact boundary and energy constraints. The answer to Question~1.8 thus distinguishes the five basic approximation properties from the stronger topological and exceptional-set conclusions. Applications include constrained integral functionals, strict $BV$ convergence of superlevel sets, nonlinear flux convergence, and stability of persistence diagrams under tameness assumptions.

We study boundary regularity for the fully nonlinear singular elliptic equation of the form $|Du|^\gamma F(D^2u)=f$, where $-1<\gamma<0$. First, we prove pointwise boundary $C^{1,\alpha}$ estimates with pointwise $C^{1,\alpha}$ Dirichlet data, which weakens the $C^2$ domain smoothness requirement from Birindelli and Demengel~\cite{BD10}. The proof uses compactness and perturbation, without flattening the boundary. Second, we extend the $W^{2,\delta}$ estimates of Li and Li~\cite{LiLi17} from balls to $C^{1,\alpha}$ domains with $C^{1,\alpha}$ Dirichlet data{, for $0<\delta\le\delta_0<1$. Here $\delta_0$ is an exponent in the interior $W^{2,\delta_0}$ estimate}. The proof combines pointwise boundary $C^{1,\alpha}$ estimates, interior $W^{2,\delta_0}$ estimates, and a Whitney decomposition.

For $1<p<2$ we prove weak and strong unique continuation properties for the solutions to $\Delta_pu+f(u)=0$ in a planar domain, with $f$ continuous and such that $|f(s)|\leq C|s|^{p-1}$: a solution vanishing on an open set vanishes identically, and so does a solution vanishing to infinite order at a single point. We can therefore make progress towards a proof of some long-standing nonexistence results available only for $p=2$, such as a celebrated one of Esteban and Lions, establishing here that for the critical $p$-Laplace equation with zero Dirichlet boundary condition on a half-plane, there are no nontrivial finite energy solutions. In the plane, unique continuation thus provides the missing ingredient for a generalisation to the $p$-Laplacian operator of a classical result of Struwe on the bubble-profile decomposition of possibly sign-changing Palais-Smale sequences associated to the Brezis-Nirenberg problem, which remains open in dimension $N\geq3$ for $p\neq2$. We obtain a characterisation of their loss of compactness in terms of the finite energy solutions of $\Delta_pu+|u|^{p^*-2}u=0$ in $\mathbb R^2$.

We prove nonlinear stability of positive nonconstant steady states for a chemotaxis–consumption system with nutrient-dependent motility and a fixed boundary nutrient concentration $b$. The motility satisfies two monotonicity conditions met by power laws and saturating functions. For each prescribed population mass, sufficiently large nutrient diffusivity $D$ yields a unique positive steady state. Small compatible perturbations generate unique global solutions converging exponentially to the steady state with their conserved mass. The result holds on general bounded domains in two and three dimensions, with a lower-regularity version on an interval. At order $D^{-1}$, the nutrient deficit is the torsion function multiplied by the mean population density and boundary nutrient value; the population contrast also depends on $\kappa_b=b\phi'(b)/\phi(b)$. Computations illustrate relaxation from compatible data and convergence to different steady states when the initial mass changes.

We prove polynomial-in-time Sobolev bounds on the approximation error between the twisted discrete nonlinear Schrödinger equation and its magnetic continuum limit. The analysis exploits complete integrability of the continuum flow through its Birkhoff representation to control approximation by a non-integrable Hamiltonian finite-difference lattice. We establish a focusing-defocusing dichotomy in uniform polynomial control: the bounds hold for arbitrary defocusing data and focusing data of sufficiently small mass, while modulational instability and aliasing yield an obstruction for large focusing data. The accompanying algebraic spatial rates are generically optimal in the Baire-category sense. The proof combines Birkhoff coordinates with modified energies for discrete Sobolev growth, while Fourier filtering recovers uniform spacetime estimates below the energy space despite degenerate lattice dispersion.

We give a dynamical characterization of the Chandrasekhar mass for cold white dwarfs governed by the radial Euler–Poisson equations with the exact Chandrasekhar equation of state. This mass is the infimum of masses for which a finite-energy classical solution with a physical vacuum can contract its entire support to a point in finite time. Below it, conservation of energy prevents complete collapse. For every mass in a sufficiently small interval above it, we construct a solution whose support shrinks to the origin and whose density concentrates there with the full mass of the star. The interior approaches a collapsing Goldreich–Weber profile. Near the vacuum boundary, a different rescaling gives a limiting profile that connects the high- and low-density regimes of the pressure law. We determine this profile and show that, after adding the electron rest energy, the conserved energy equals that of the Goldreich–Weber reference solution. The proof combines matched asymptotic expansions with local existence and energy estimates that extend solutions backward from times approaching collapse to a common earlier time.

Motivated by a class of three-component Fourier ODE systems consisting of a conserved mode coupled to a pair of oscillatory modes, we develop a "potential-vorticity-wave" (PV-Wave) decomposition strategy for constructing adapted energy variables in the stability analysis of Couette flows. The strategy separates the conserved potential-vorticity component, symmetrizes the remaining wave subsystem, and introduces an additional correction to control the couplings generated by time-dependent Fourier coefficients. We apply this framework to study the linearized dynamics of 3D Boussinesq MHD system and Rotating Boussinesq system for Couette flows and quantify the inviscid damping and parameter-dependent amplification arising from the interaction of Alfvén-gravity waves and inertial-gravity waves.

Since the pioneering work of Taylor and Goldstein in 1931 laid the foundation for the study of stratified shear flows, non-parallel configurations have received comparatively little attention. In this article, we study the stability of the three-dimensional Boussinesq system near the plane Couette flow $V^*=(y,0,0)$ under the vertically stratified background $\eta^*=\alpha z$. The shear and stratification directions are perpendicular. This geometry produces two structural effects that are absent in the parallel configuration: an additional nonlocal coupling in the nonzero-mode system, which prevents the system from having a valid symmetrization structure without reformulation, and a non-parallel/wave-shear coupled lift-up mechanism for the zero modes. The $L^p$ transient growth due to the lift-up effect can be suppressed either by sufficiently strong stratification (when $p>2$), or under suitable assumptions on the initial data (when $p\geq 2$). To study the nonzero-mode system, we introduce a reformulation based on the linearized potential vorticity (PV), a fundamental quantity in geophysical fluid dynamics, to recover a better energy structure. We establish the inviscid damping of $u_{1,\neq}$ and $u_{2,\neq}$. We further apply this method to study the linear stability of the tilted Couette flow. Based on the new formulation through PV, we also investigate the nonlinear stability of the plane Couette flow for Sobolev perturbations on $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. We consider initial data with finite weighted Sobolev norm, where the weight is chosen to suppress the lift-up effect at the linear level, so that transient growth arises only through nonlinear interactions and the nonlinear lift-up can be suppressed by strong stratification.

We study the growth of the spectral function of the Laplace-Beltrami operator on a Riemannian manifold $M$ of dimension $n$. For $n\geq 3$ we show that for any $x\in M$ there is a full measure subset $Z_x\subset M$ such that for all $y\in Z_x$ \[E_{\Lambda}(x,y) = O_{M,x,y,\varepsilon}(\Lambda^{n/2}(\log\Lambda)^{3/2}(\log\log\Lambda)^{1/2+\varepsilon}). \] For $n=2$ we prove a stronger statement. For any $x\in M$ there is a full measure subset $Z_x\subset M$ such that for all $y\in Z_x$ \[E_{\Lambda}(x,y) = O_{M,x,y,\varepsilon}(\Lambda^{5/6}(\log\Lambda)^{1/6}(\log\log\Lambda)^{1/6+\varepsilon}) .\]

We demonstrate a modification of the high-low method of Guth-Maldague-Wang that yields a decoupling theorem which gives the sharp estimate for the discrete restriction constant for the parabola and also a diameter-free bound for the three-fold additive energy of rational points on the parabola and circle. These results were recently obtained by Deng-Fan-Guo-Guo-Luo and Cushman-Demeter-Wu, respectively. The novel features in this argument are an orthogonal expansion of $F^3$ and a new square function that also keeps the wavepackets discarded after pruning, resulting in new and more efficient high and low lemmas.