Papers

We prove local rigidity of every smooth conformally Euclidean metric for the anisotropic Calderón problem on smooth compact domains $M\subset\mathbb R^n$, $n\ge3$. Any smooth Riemannian metric $g$ sufficiently close to a fixed background $g_0=e^{2c}\mathsf e$ in $H^s(M)$, with integer $s>n/2+1$, having the same Dirichlet-to-Neumann map satisfies $g=\Phi^*g_0$ for a smooth diffeomorphism $\Phi$ fixing the boundary pointwise.

We study the Einstein-Vlasov system near the Schwarzschild spacetime in spherical symmetry. In particular, we consider the case when the matter is supported on bounded geodesics. In this regime, the decay of the matter is driven by phase mixing rather than dispersion. For the linearized problem, we obtain quantitative phase mixing; the key new ingredient proved here is a monotonicity property of the period function. For the nonlinear coupled problem, we prove phase mixing up to the timescale expected from the nonlinear echo mechanism; this is achieved by analysis on dynamical action angle variables and the integral estimates with the vector field method. This construction is made possible by our gauge choice, an isotropic coordinate system together with a foliation of prescribed mean curvature, in which the Einstein equations gain derivatives over the matter due to its elliptic nature.

We prove nondegeneracy, modulo horizontal translation, of the normalized Sadovskii vortex patch. Building on the free-boundary and root-map framework developed by Huang and Tong, we first obtain a precise quantitative approximation of an exact Sadovskii profile, trapping its free boundary between two explicit barriers constructed around a rational approximation. This localization is then used to control the natural weighted quadratic form at the touching configuration. In the odd class, a ground-state identity shows that horizontal translation is the only zero mode. In the even class, dilation gives a negative direction, while a finite-rank comparison on a compact part of the boundary together with analytic contact-tail estimates yields a positive gap on a codimension-one subspace. Consequently, the Sadovskii dipole is locally rigid at the linearized level once translation is fixed. The computer-assisted part is confined to rigorous interval verification of a finite collection of explicit inequalities; the continuum, contact and operator reductions are proved analytically.

We formulate and prove existence of pure quasifree Hadamard states for the Klein–Gordon field with Dirichlet boundary condition on a globally hyperbolic spacetime with timelike boundary. The microlocal condition is imposed on the one-particle Hilbert-space-valued distribution $p\circ G_D$ rather than on the two-point function. This is expressed by a $b$-wave-front set condition on the compressed future cone. The analysis on the manifold with corners $\bar M\times\bar M$ is then almost completely avoided. We prove that our version of the Hadamard condition is universal in the class of quasi-free Hadamard states. We also prove a propagation theorem for positive-energy microlocal splittings which uses only a microlocal splitting in the small $b$-calculus and well-posedness of the Dirichlet problem, and in particular requires no propagation theorem for reflected or gliding rays; combined with an explicit computation in the ultrastatic case and a Fulling–Narcowich–Wald deformation argument this yields the construction of a pure quasi-free Dirichlet–Hadamard state.

Let $2\leq k<n/2$ and let $(M^n,[g])$ be a smooth, closed, connected, and locally conformally flat Riemannian manifold with a $k$-admissible metric in the conformal class $[g]$. We prove a stability result of the $\sigma_k$-curvature inequality on $M$, in the sense that if equality is almost satisfied for some conformal metric, then its conformal factor is close to a minimizer of the inequality. Closeness is measured quantitatively in terms of Sobolev norms of the conformal factor, namely with respect to the $W^{1,2}$- and the $W^{1,2k}$-norm. In the non-degenerate case, these norms come with optimal exponents $2$ and $2k$, respectively, whereas in general the exponents are $2+\gamma$ and $\max\{2k,2+\gamma\}$ for some $\gamma\geq 0$ originating from a Łojasiewicz inequality. This extends a previous result by Frank and the author from $k=2$ and the sphere to $2\leq k<n/2$ and the full class of manifolds originally considered by Viaclovsky. It also extends a previous result by Engelstein–Neumayer–Spolaor from $k=1$ to the setting of fully non-linear scalar curvatures.

In this short note we prove that, given a $\mathbb{C}$-elliptic operator $\mathcal{A}$ and a map $u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V)$ satisfying \( \nabla_{\mathrm{ap}}u\in L^1(\Omega;V\otimes\mathbb{R}^d), \) then $u\in \mathrm{BV}(\Omega;V)$. The result is quantitative and follows from the Korn-type estimate \[ |Du|(\Omega) \leq C_{d,\mathcal{A}}\left( \|\nabla_{\mathrm{ap}}u\|_{\mathrm{L}^1(\Omega)} + |\mathcal{A}u|(\Omega) \right), \] valid for every such $u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V)$. As a consequence, we obtain the characterization \[ \mathrm{BV}^{\mathcal{A}}(\Omega;V)\setminus \mathrm{BV}(\Omega;V) = \left\{ u\in \mathrm{BV}^{\mathcal{A}}(\Omega;V): \nabla_{\mathrm{ap}}u\notin L^1(\Omega;V\otimes\mathbb{R}^d) \right\}. \] Generative AI has been exploited. The usage is detailed in a specific Section.

The Chern-Simons-Schrödinger equation (in the temporal gauge) arises as the Hamiltonian flow of the abelian Higgs energy on $\mathbb R^2$. Manton (arXiv:hep-th/9701027) introduced the equation as a model for the dynamics of the critical points of the energy, known as vortices. He conjectured that, for a certain range of coupling constants, the vortex motion under the Chern-Simons-Schrödinger flow can be effectively captured by a first-order ODE on the moduli space of self-dual vortices constructed by Jaffe-Taubes (1980) and Samols (1992). As a first step towards a rigorous proof of Manton's conjecture, we formulate the Cauchy problem in DeTurck gauge within the natural energy space and prove global well-posedness. We also obtain, as a corollary of the well-posedness theory and the stability results in our previous work (arXiv:2603.24900), orbital stability of the self-dual vortices under the Chern-Simons-Schrödinger flow near self-dual coupling. The heart of our analysis lies in developing a geometric Littlewood-Paley theory based on the covariant heat equation in caloric gauge, which we use to perform a paradifferential-style decomposition of the Chern-Simons-Schrödinger equation.

Becker et al. produced a striking trace formula for the sum of fourth powers of generalized magic angles, $ \theta $, for the chiral model of twisted bilayer graphene (TBG) with the exact Bistritzer–MacDonald potential: $\sum \theta^{4} = 8 \pi / \sqrt 3 $. The purpose of this note is to generalize this formula to a larger class of potentials satisfying the symmetries of the model.

In this paper, we give a negative answer to Wang's conjecture. For every $n\geq 3$, there exist $\varepsilon>0$ and $\delta \in (0,1)$ such that, for all $q$ and $\lambda$ satisfying \[ \frac{n}{n-2}-\varepsilon<q\le \frac{n}{n-2}, \qquad \frac{1}{q-1}\cdot \delta<\lambda\le \frac{1}{q-1}, \] there exists a bounded Euclidean domain \(\Omega\subset\mathbb{R}^n\) with principal curvatures bigger than \(1\) for which the following nonlinear Robin problem admits a nonconstant positive solution: \begin{align*} \begin{cases} \Delta u=0, & \text{in}\ \Omega,\\[2mm] \dfrac{\partial u}{\partial\nu}+\lambda u=u^q, & \text{on}\ \partial\Omega. \end{cases} \end{align*}

We establish one form of the vortex filament conjecture for the three-dimensional Gross-Pitaevskii equation. Given any smooth closed embedded binormal flow of curves on a compact time interval, we construct, in the small-core limit $\varepsilon\to0$, a family of exact solutions whose degree-one vortex filaments converge uniformly to the prescribed flow. Near the filament the solutions have the standard planar vortex profile, while away from it their phase gradients converge to the associated Biot-Savart field. We also derive a refined modulation law for the vortex filament.

We describe how strong vorticity fluctuations appear in three-dimensional incompressible turbulence driven by a large-scale random force. By combining analytical and numerical methods, we construct a vorticity instanton: the most probable force history driving the flow to a prescribed large vorticity at a given point and time. We show how the smooth, large-scale force produces a long vortex filament with a thin, viscous-scale core. Our main finding is that the standard mechanism of amplifying a vortex tube through radial compression and axial stretching is only effective up to a threshold value. Producing vorticity above the threshold requires an additional ingredient: vorticity waves propagating along the tube. The final vorticity is the result of two colliding waves. This mechanism has striking similarities to the one recently suggested by OpenAI for creating infinite vorticity in finite-time Navier–Stokes blow-up.

In this paper, We obtain dispersive estimates for solutions to the Schrödinger equation with a uniform magnetic field on the hyperbolic plane \(\mathbb{H}\). The key ingredient is an explicit representation formula for the kernel of the corresponding Schrödinger propagator. As a consequence, we prove the corresponding Strichartz estimates for all admissible pairs on \(\mathbb{H}\).

We prove pointwise decay and scattering for small-mass solutions to the focusing and defocusing continuum Calogero–Moser models under suitable decay assumptions on the initial data. Our proof is based on an explicit formula for solutions and the exact conservation of the Galilean vector field associated to the equation.

In this paper, we study an inverse boundary value problem for a generalized Jordan-Moore-Gibson-Thompson equation with Westervelt-type nonlinearity and space-dependent coefficients. This third-order (in time) hyperbolic equation models nonlinear ultrasound propagation in viscous and thermally relaxing media. Assuming that the diffusivity and the sound speed are known a priori, we investigate the simultaneous stable determination of the friction coefficient, the weak damping coefficient, the potential, and the nonlinear coefficient from the associated Dirichlet-to-Neumann map. The proof combines the finite-difference linearization method with the construction of Gaussian beam and geometric optics solutions. These arguments reduce the inverse problem to stability estimates for (attenuated) geodesic ray transforms under the foliation condition. As a consequence, we obtain Hölder-type stability estimates of recovering the linear and nonlinear coefficients.

This paper presents two types of novel, qualitative and quantitative, estimates for positive solutions of general uniformly elliptic PDEs in divergence form. First, we prove a Morel-Oswald type of extension of the Hopf-Oleinik lemma, in which in addition we specify the sharp dependence of the constant in the data of the operator and the size of the domain. Second, we establish a new differential Harnack (logarithmic gradient) estimate for non-homogeneous equations, as well as an optimal global differential Harnack estimate for homogeneous equations.

We study the fractional porous medium equation on complete noncompact Riemannian manifolds with nonnegative Ricci curvature for $s\in(0,1]$ and $n>2s$. Assuming that the manifold is $s$-nonparabolic, so that the fractional Laplacian admits a suitable positive minimal Green function, we use this Green function to introduce a natural weighted space of initial data, strictly larger than $L^1$. This leads to a weak dual, or potential, formulation of the equation, for which we prove existence for nonnegative initial data in the weighted space. We then establish quantitative local smoothing estimates for initial data in either $L^1$ or the Green-weighted space. Under additional noncollapsing and uniform volume-growth assumptions, we obtain global smoothing estimates. We also show that, even in $\mathbb{R}^n$, data in the Green-weighted space need not generate bounded solutions unless a suitable uniform weighted integrability condition is imposed. The short- and long-time behaviors predicted by our estimates are shown to be optimal in appropriate senses. When $s\in(0,1/2)$, the results require only $\operatorname{Ric}\geq0$, together with a noncollapsing assumption where needed. Our estimates also cover the case $s=1$, several of them being new in that setting. Finally, we extend the approach to more general filtration equations.

We establish a priori interior $C^{1,\alpha}$ and $C^{2,\alpha}$ Schauder estimates for a class of $Q$-valued quasilinear elliptic systems which are perturbations of the Laplace system, for arbitrary multiplicity $Q$, domain dimension $n\geq2$, and target dimension $k$. The $C^{1,\alpha}$ estimate generalises the work of Simon and Wickramasekera in \cite{SW16} on $2$-valued solutions of linear systems and concerns weak solutions of divergence-form systems, while the $C^{2,\alpha}$ estimate concerns strong solutions. In dimension $n=2$, both estimates hold for every $0<\alpha<1/Q$. In dimensions $n\geq3$, there exists $\delta=\delta(n,k,Q)>0$ such that both estimates hold for every $0<\alpha<\delta$. As applications, we obtain a small-slope Schauder estimate and a small-slope Bernstein theorem for $Q$-valued maps whose graph varifolds are stationary. Combining the Schauder estimate with the recent branch-set stratification theory of Krummel–Minter–Wickramasekera in \cite{KMW26}, we further prove that, for every $\gamma>0$, the branch set $\mathcal B_u$ of any $C^{1,\gamma}$ $Q$-valued map $u$ whose graph varifold is stationary satisfies $\dim_{\mathcal H}\mathcal B_u\leq n-2$. This bound is sharp already in codimension one.

We extend reflection coupling for uniform-in-time propagation of chaos from Euclidean space to complete Riemannian manifolds. Intrinsic reflection along minimizing geodesics replaces the Euclidean difference process, and the associated endpoint-index term quantifies the competition between curvature, confinement, and interaction. Under an endpoint-index bound and a radial Laplacian bound, we establish global well-posedness, uniform moment estimates, and uniform-in-time propagation of chaos with rate $N^{-1/2}$ in a modified Wasserstein distance. We further obtain exponential contraction of the nonlinear McKean–Vlasov semigroup, uniqueness of its invariant probability measure among probability measures with finite second moment, and exponential convergence to equilibrium. Although a Ricci lower bound is an important special case, our framework also includes a curvature-spike class with $\inf_M\operatorname{Ric}=-\infty$.

In this paper, we study two families of quasilinear equations: Hunter-Saxton type and Camassa-Hom type equations, with a paramerter $\lambda\in(0,1)$ whose solutions form cusp singularities. When $\lambda=1$, the first system becomes the scalar conservation law. The main result of this paper is to give regularity of two types of structurally stable singularities: Type I on the singular curve, Type II at the point where cusp singularity forms, for some $\lambda\in(0,1)$. When $\lambda\rightarrow 1$, our result indicts the $C^{1/3}$ regularity at the point where singularity forms, which agrees with the regularity of the generic pre-shock solution.

We study the momentum equation of the decoupled unregularised Hibler sea-ice model with non-negative ice mass and ice strength, both of which may vanish. The ocean drag provides coercivity, while the stress law and mixed boundary conditions are encoded by a convex dissipation functional. We prove existence, uniqueness and stability for the time-discrete problem and reconstruct an admissible stress. A Rothe approximation yields global weak variational solutions for time-dependent coefficients under a one-sided growth condition on the mass. Weak solutions are unique when the ice strength is independent of time. For spatially Lipschitz ice strength, finite dissipation also yields a local measure structure of the weighted deformation and a global bound on the negative part of the weighted divergence.

In this paper, we present the space-time derivative estimates of solutions to the MHD equations. It is a generalization of a priori bounds for the classical Navier-Stokes system by Duff in \cite[Acta Math. 164, 1990]{[Duff]} and gives an affirmative answer to a question proposed by Zheligovsky in \cite[Mathematics. 9, 2021]{[Zheligovsky]}. In addition, we show that $u, B \in L^{\f{q}{q-3}}(0,T;L^{q} (\mathbb{R}^{3}))$ with $6\leq q\leq\infty $ and $ D ^{k}u, D ^{k}B \in L^{\f{q}{ q(k+1)-3}}(0,T;L^{q} (\mathbb{R}^{3}))$ for $k\geq1, 2\leq q\leq\infty$ in this system.

In this work, we consider relative overdetermined problems for the first eigenfunction of the Laplacian for domains in cones, and the related question of minimizing the first eigenvalue among sets of a given fixed measure. By means of a shape derivative analysis, we show that the spherical sector is a critical shape and obtain a geometric condition on the cone for its stability/instability. By a concentration-compactness argument, we prove the existence of a minimizer, which moreover is bounded, open, connected, and whose relative boundary is regular almost everywhere. By another domain variation argument, we conclude that the minimizers admit a solution for the overdetermined problem.

In this paper, we are concerned with space-time derivative estimates of solutions to the fractional Navier-Stokes equations. It is shown that $\Lambda^{n\alpha}u^{(m)}_{t} \in L^{\frac{2(6\alpha-5)}{4m\alpha+2n\alpha+4 \alpha-5}}~~~~~~~(0,T; L^{2}(\mathbb{R}^{3}))$ and $ \Lambda^{n }u^{(m)}_{t} \in L^{\frac{2(6\alpha-5)}{4m\alpha+2n +4 \alpha-5}}~~~~~(0,T; L^{2}(\mathbb{R}^{3}))$. This generalizes a priori bounds for the classical Navier-Stokes system by Duff in [7, Acta Math. 164, 1990] and Boutros and Gibbon's spatial derivative estimates in [1, Nonlinearity 37, 2024]. In addition, we derive that $ u \in L^{\frac{q}{q-3}}~~(0,T;L^{q} (\mathbb{R}^{3}))$ with $ 6\leq q\leq\infty $ and $\Lambda^{k}u \in L^{\frac{q}{ q(k+1)-3}}~~~(0,T;L^{q} (\mathbb{R}^{3})) $ with $k\geq1, 2\leq q\leq\infty $ in the standard Navier-Stokes equations.

In this paper, we study the nonlinear boundary value problem \begin{equation*} \begin{cases} -\Delta u=e^{2u},& \mbox{in } {\mathbb{D}},\\ \frac{\partial u}{\partial\nu}+\lambda=e^u ,& \mbox{on } {\mathbb{S}^{1}}, \end{cases} \end{equation*} where $\mathbb D$ is the unit disk, $\lambda$ is a constant and $\nu$ denotes the outer unit normal on $\mathbb S^1$. For $0<\lambda\le2$, we establish a complete classification of smooth solutions. For $2<\lambda\leq3$, we prove a dichotomy between radial solutions and nonradial solutions. We develop a Hardy-Wronskian boundary rigidity method that transforms the nonlinear boundary value problem into a spectral rigidity problem for normalized holomorphic frames. This method exploits the complex-analytic structure of the Liouville equation with a Robin boundary condition and provides a new rigidity framework for related two-dimensional elliptic boundary value problems.

We study weakly interacting bosons on the three-dimensional torus at temperatures proportional to the critical temperature for Bose–Einstein condensation. We show that, as the expected particle number tends to infinity, the grand canonical Gibbs state is asymptotically described by the coherent state quantization of a random field that converges to a novel Bogoliubov random field. Moreover, the point process associated with the Gibbs state can be approximated by a Cox process. These results are based on and extend the approximation of the Gibbs state recently obtained by the first two authors and Nam. Using these approximations, we compute limiting distributions of the number of particles outside the condensate, the joint occupation statistics of finitely many momentum modes, and particle-number fluctuations in macroscopic subsets of the torus, all governed by the Bogoliubov field. We further prove convergence of the empirical measure of the associated point process to the uniform measure and convergence of the microscopic particle-number statistics to the boson point process. The latter result establishes the universality of particle-number statistics on microscopic length scales.

Let $0<\alpha<2$ and $p=2/(2-\alpha)$. We study critical nonlinear convolution inequalities for kernels $K:\mathbb{R}^2\setminus\{0\}\to\mathbb{R}^m$ and functions $\Phi:\mathbb{R}^m\to\mathbb{R}$ positively homogeneous of degrees $\alpha-2$ and $p$. We assume angular Lipschitz regularity for $\Phi$ and angular $C^{0,\gamma}$ regularity for $K$, where $0<\gamma\le1$ and $\gamma=1$ if $p<2$. On bounded finitely cornered planar domains with a common Dini tangent modulus on the boundary branches, we characterize $|\int_\Omega\Phi(K*f) dx|\le C\|f\|_{L^1}^p$ for real densities $f\in L^1(\mathbb{R}^2)\cap L^s_{\mathrm{loc}}(\mathbb{R}^2)$, $1<s\le\infty$, by signed angular cancellation on the plane, tangent half-planes, and full vertex tangent cones. An analogous criterion holds on infinite sectors for compactly supported mean-zero densities. For the Newton kernel and each nonzero angularly Lipschitz, positively two-homogeneous $\Phi:\mathbb{R}^2\to\mathbb{R}$ with vanishing half-circle moments, we construct a bounded convex $C^1$ domain $\Omega_\Phi$, independent of $\varepsilon$, and smooth tests $u_\varepsilon$ satisfying $\|\Delta u_\varepsilon\|_{L^1(\Omega_\Phi)}+\|\partial_nu_\varepsilon\|_{L^1(\partial\Omega_\Phi)}\le2$, while, despite tangent cancellation, their nonlinear integrals grow in magnitude like $\log\log(1/\varepsilon)$. On the Dini class, a constant-preserving linear extension with compact support modulo the domain mean and controlled measure-valued Laplacian yields the same necessary-and-sufficient criterion for the Newton-kernel PDE inequality, without exact corner charts. We also construct infinite-dimensional cancellation spaces containing nonquadratic functions and classify the quadratic locus in polygon moduli.

We investigate forward and backward smoothing effects in Lebesgue spaces $L^p$ and $\mathcal{M}^p:=L^{p,\infty}$ for the Cauchy problem associated to the nonlinear and nonlocal fractional diffusion equation $\partial_t u+(-\Delta)^{\frac\sigma2}|u|^{m-1}u=0$ in $\mathbb{R}^N$, $0<\sigma<2$, in the very fast range $0<m\le m_c:=\frac{N-\sigma}{N}$. We prove that very weak solutions have an $\mathcal{M}^p$–$L^\infty$ smoothing effect if $p>p^*:=\frac{N}{\sigma}(1-m)$, and we construct counterexamples showing the failure of any $L^p$–$\mathcal{M}^q$ forward ($q>p$) smoothing effect if $1< p< p^*$, $m<m_c$ or $L^1$–$\mathcal{M}^q$ ($q>1$) if $m=m_c$. We also prove a backward $\mathcal{M}^p$–$L^1$ smoothing effect whenever $1\le p<p^*$, $m<m_c$, and we construct counterexamples showing that there is no $L^p$–$\mathcal{M}^q$ backward ($q<p$) smoothing effect if $p> p^*$. Regarding the threshold value $p=p^*$, we prove that all solutions starting in $\mathcal{M}^{p^*}$ become extinct in finite time, and show the failure of any $\mathcal{M}^{p^*}$–$\mathcal{M}^q$ smoothing before extinction for any $q\neq p^*$. The construction of counterexamples is based on new uniqueness and comparison results for very weak solutions, combined with the existence of self-similar solutions with suitable properties. The same approach yields new counterexamples for both forward and backward smoothing effects also in the local case $\sigma=2$.

In this paper, we investigate the Lipschitz stability of the master field and its functional derivative with respect to the measure variable for the master equation in mean field games. A key novelty of this work is the derivation of a nonlocal parabolic partial differential equation governing the functional measure derivative along a characteristic measure flow. Based on this equation, we establish a Carleman estimate for the difference of the measure derivatives associated with two sufficiently regular classical solutions. We also establish a Carleman estimate for the difference of the corresponding master fields, yielding a terminal-to-interior Lipschitz stability estimate along a characteristic measure flow. Together with stability of the associated measure flows, these estimates are used to control the resulting source terms. We then obtain a terminal-to-interior Lipschitz stability estimate for both the master field and its functional measure derivative in terms of the discrepancies of the terminal data. The result provides quantitative control of the master field with measure-variable regularity, and develops a Carleman-based framework for the stability analysis of master fields and their measure derivatives in mean field game master equations.

We construct compact, exponentially stable $C_0$-semigroups $S$ on Hilbert spaces $X$ such that, for every $T>0$, the stochastic convolution $U_g(t)=\int_0^t S(t-s)g(s)\,\mathrm{d}\beta_s $ is unbounded on $[0,T]$ with positive probability for some predictable $g\in L^\infty(\Omega;L^2(0,T;X))$, where $\beta$ is a real Brownian motion. One example is even an analytic semigroup. For the other, the negative generator $A$ has sectorial angle $\pi/2$ and a bounded $H^\infty$-calculus. Our main tool is a necessary condition: for exponentially stable semigroups, an $L^2$-maximal estimate on $\mathbb R_+$ forces a lower square-function estimate for the negative generator. We deduce this implication from a new identity involving the stochastic convolution, the subordinated Poisson semigroup, and a stopped Brownian motion. Combining this condition with Schauder multipliers on a conditional trigonometric basis yields counterexamples to the maximal estimate. An extrapolation argument then produces the integrands with unbounded stochastic convolutions. Finally, an energy-adapted chaining argument gives continuity and maximal estimates for arbitrary $C_0$-semigroups under the stronger condition $g\in L^2([0,T]; L^\infty(\Omega;X))$.

This work is a mathematical study of the multiphasic formulation of the Vlasov equation, which consists in recasting this collisionless kinetic equation as a system of coupled pressureless Euler equations. This framework allows to consider solutions that are only measure-valued in the velocity variable and is thus relevant to tackle physical problems where rough velocity distributions, such as Dirac masses, naturally arise. We specifically address the case of nonlinear Vlasov equations where the force field is one derivative more regular than some moments in velocity of the solution. Under this key assumption, a unified theory of local well-posedness at finite regularity (typically in Sobolev spaces) is developed from scratch for the multiphasic formulation, yielding corresponding results for the Cauchy problem of the associated Vlasov equation at low regularity. We thoroughly apply this abstract theory to two classes of equations, namely Vlasov-Poisson type systems, and Vlasov-Navier-Stokes type systems. In addition to the justification of the monokinetic limit, it also leads to specific applications for each class of equations, allowing possibly rough velocity distributions. For Vlasov-Poisson type systems, we justify the semiclassical limit from Hartree equations, and we describe the nonlinear instability of homogeneous equilibria. For Vlasov-Navier-Stokes type systems, we establish nonlinear asymptotic stability near monokinetic profiles. New results allowing a separation between the regularity in space and in velocity are proven, and along the way, we also provide new proofs of known results and generalize them to the case of rough solutions. Finally, the flexibility of the framework is exploited to obtain extensions to more sophisticated systems such as the Vlasov-Poisson equation for ions, or the Vlasov equation coupled with the compressible Navier-Stokes system.