Papers

We study the focusing inhomogeneous nonlinear Schrödinger equation with an inverse-square potential $$ i\partial_t u+\Delta u-a|x|^{-2}u+|x|^{-b}|u|^{\alpha}u=0, \qquad (t,x)\in\mathbb{R}\times\mathbb{R}^{N}, $$ in the mass-supercritical and energy-subcritical regime $$ \frac{4-2b}{N}<\alpha<\frac{4-2b}{N-2}, $$ where $N\ge3$ and $0<b<2$. Two main lines of research have developed largely independently for this model: scattering theory below the ground-state threshold and the variational analysis of solutions with prescribed mass. In this work, we establish a connection between these two approaches.

We consider an abstract wave-type evolution equation with a propagation speed depending only on time. We assume that this propagation speed is the product of a monotone, but possibly degenerate, shape function and an oscillating factor bounded from above and from below by positive constants. We investigate how degeneracy and oscillations interact in determining the derivative loss of solutions. After reducing the problem to a family of ordinary differential equations through spectral analysis, we derive frequency-dependent energy estimates. In the purely degenerate case, monotonicity of the shape function prevents excessive loss, and solutions lose at most one derivative. In the presence of oscillations, the resulting derivative loss is described by a minimization problem in which the degenerate and oscillatory effects genuinely interact. The proof is based on a decomposition of the time interval into frequency-dependent regions, where different adapted energies are used.

We consider the existence of nonradial sign-changing solutions to the problem $\Delta u-\mu u+|u|^4u=0$ with zero Neumann boundary conditions in the unit ball for an arbitrary fixed parameter $\mu>0$. We prove that, for every $\mu>0$, this problem admits infinitely many distinct nonradial sign-changing solutions. More precisely, for every sufficiently large even integer $K$, we construct a solution by gluing $K$ symmetrically arranged, suitably transformed copies of a sign-changing entire solution of the critical equation in $\mathbb{R}^3$. The construction combines an inner–outer gluing scheme with a finite-dimensional reduction. A detailed expansion of the reduced energy allows us to locate an interior critical point in the admissible parameter region and thereby complete the construction.

The filtering problem asks to estimate a signal process, the initial state of which is unknown, given its partial and noisy observations. From a probabilistic perspective, the best estimate at each time is the optimal filter, which is the conditional law of the signal given all observations up to the time. The problem of asymptotic stability of the optimal filter, which asks whether the filter initialized with the incorrect initial law converges to the correctly initialized filter, is of fundamental practical and theoretical importance. This paper studies the quantitative asymptotic stability of the optimal filter for signal-observation models with linear drift and additive Brownian noise. When the initial law is additionally assumed to be Gaussian, the optimal filter reduces to the celebrated Kalman-Bucy filter, for which the detectability condition on the signal and observation matrices is the sharp condition ensuring asymptotic stability. In the linear setting where the detectability condition holds but the initial law need not be Gaussian, we make the following contributions by studying the Zakai equation satisfied by the unnormalized filter. First, we study the asymptotic behavior of the deterministic part of the Zakai equation in the general case with nonlinear drift, and place the detectability condition in this context. Second, in the case of linear drift, we characterize the solution of the Zakai equation (hence the optimal filter) by a gauge transformation that separates the finite-dimensional stochastic dynamics from the infinite-dimensional deterministic dynamics. Third, we use this representation to establish asymptotic exponential filter stability in the $\chi^p$-divergence (with respect to a reference optimal filter), the total variation norm, and the $2$-Wasserstein distance.

Under mild mass or energy assumptions, we prove uniform $L^\infty$ bounds for all distributional solutions of a multi-dimensional system of nonlocal semilinear equations. The setting comprises Ginzburg-Landau systems and Allen-Cahn equations, and the results obtained are new even for scalar equations.

Let $w_0$ be the positive homoclinic solution of $-w''+w=w^p$ and set $c_p=w_0(0)$. We study bounded positive solutions of \[ -\Delta v+v=v^p\quad\text{in }\mathbb R^N_+, \qquad v=c_p\quad\text{on }\partial\mathbb R^N_+, \qquad v(x',x_N)\to0 \] uniformly as $x_N\to\infty$. This is the threshold boundary value left open by Fernández and Weth [Math.\ Ann.\ 383 (2022), 361–397]. We prove a dimension-dependent rigidity/nonuniqueness picture. For every $p>1$, the profile $w_0(x_N)$ is the unique bounded positive solution when $2\le N\le5$, whereas for $N\ge8$ there is a one-parameter family of bounded positive non-one-dimensional solutions. In the critical dimension $N=7$, nonuniqueness holds whenever a cubic coefficient $\kappa(p)$ of the exact reduced nonlinearity is positive; in particular $\kappa(p)>0$ for all $p\ge2$, and continuity extends this range below $2$. The corresponding amplitudes have Fowler ends, with their limiting energy, neck size, bulge size, and logarithmic period determined to leading order. The case $N=6$, as well as the complementary range in $N=7$, remain open. The proof combines a boundary-adapted Modica estimate, projection onto the kernel of the one-dimensional linearized operator, an automatic tail non-concentration estimate, and a nonlinear cell problem yielding the exact reduced nonlinearity. For $N\ge8$ the reduced linearization is invertible in weighted spaces. In $N=7$ the critical resonance is bypassed by solving the projected radial equation nonlinearly and closing the transverse equation by a Schauder–Tychonoff fixed point.

We study $-\Delta_p u = |u|^{q-1}u$ in $\mathbb{R}^N$ with $N>p\ge 2$, for solutions that are stable outside a compact set, with no assumption of sign, boundedness or symmetry. Damascelli, Farina, Sciunzi and Valdinoci settled the subcritical range for $p>2$, and treated the supercritical range $p^*-1<q<q_c(N,p)$ only for radial solutions, stating that without radial symmetry they were not able to conclude. The obstruction is that the arguments available for $p=2$ rest on a monotonicity formula with no monotone analogue when $p\ne2$. We remove the radial hypothesis: every $C^1$ weak solution stable outside a compact set is trivial when $p^*-1<q<q_c(N,p)$. The whole range $p-1<q<q_c(N,p)$ with $q\ne p^*-1$ is thereby settled, as it is for $p=2$ by Farina's theorem, and every finite Morse index solution in it is trivial. In its place we iterate the comparison principle against an explicit two-parameter supersolution, upgrading the decay forced by stability to that of the $p$-harmonic fundamental solution; this step uses no stability, only smallness of the self-similar decay, and is stated on its own. The upper threshold is sharp: for every $q>q_c(N,p)$ we exhibit positive bounded radial solutions stable outside a compact set. At the Sobolev endpoint, every $C^1$ solution stable outside a compact set has finite energy; for $p>2$ this removes the a priori $\mathcal D^{1,p}$ assumption from the low-Morse-index classification of Farina, Mercuri and Willem and answers their question on the Morse index of the Aubin–Talenti extremals: it is one. At the upper endpoint, $q_c(N,p)$ is the exact threshold for a nonzero homogeneous weak solution on $\mathbb R^N\setminus\{0\}$ to be stable outside a ball, and at the threshold there are exactly two.

In this paper, we consider the sixth-order Boussinesq equation with a cubic nonlinearity, \[ u_{tt}-u_{xx}+k u_{xxxx}-u_{xxxxxx}+\left( u^3 \right) _{xx}=0, \ \ \mbox{with }k=\pm 1,\] posed on both $\mathbb{R}^+$ and $\mathbb{R}$. The local wellposedness are improved in related Bourgain-type spaces, $X^{s,b}$, for $s>-\frac12$ with $b=\frac12$ and $b>\frac12$ accordingly (See. \cite{esfahani2025well,zhong2024local}). The key ingredient is to refine the trilinear estimate on Burgain type space by adapting ideas from Tao's multiplier method, interpolation theory and our early work on IBVPs of dispersive equations \cite{li2020low,tao2001multilinear}.

In this paper, we continue the study of Himonas and Yan\cite{himonas2024schrodinger,himonas2026higher,himonas2024higher} on the nonlinear Schrödinger equation with a dispersion of order $2m$ and cubic nonlinearities, \[ iu_t+\left( -1 \right) ^{m+1}\partial _{x}^{2m}u=N_k\left( u,u,u \right), \] where, $m\ge1$ being an integer, \[N_0\left( u,u,u \right) =uuu,\quad N_1\left( u,u,u \right) =\bar{u}uu,\quad N_2\left( u,u,u \right) =\bar{u}\bar{u}u,\quad N_3\left( u,u,u \right) =\bar{u}\bar{u}\bar{u},\] We derive trilinear estimates at lower regularity and thereby prove that the cNLS-2m on the half-line is well-posed at the optimal regularity $s=-\fr{m-1}{2}$. This improves the previous result, $s>-\fr{m-1}{2}$, and answers an open question left in \cite{himonas2026higher}. In addition, for $k=0,2$, we establish well-posedness at $s=-\fr{m-1}{2}$ as well. Moreover, for $k=3$, the nonlinearity exhibits a stronger resonance relation, which implies well-posedness for $s>-\fr{2m-1}{3}$. Our derivation of the trilinear estimates relies on varieties of Strichartz estimates, which differs from the $[k;Z]$-multiplier norm method employed in \cite{himonas2026higher}.

We study large-data continuation below the critical energy for the two-dimensional sphere-valued Schrödinger map \[ \partial_t u=u\times\Delta u, \qquad u:[0,T_+)\times\mathbb{R}^2\to\mathbb{S}^2, \qquad E(u_0)<4\pi. \] Working directly with the original equation through the complex tangential fields $W_m=\partial_m u+\mathrm{i} u\times\partial_m u$ and the ambient connections $\Gamma_\alpha$, we prove frequency-envelope bounds governed by the critical norm $\|\nabla u\|_{\ell^2L^4_{t,x}}$. As consequences, $T_+=\infty$ if $\|\nabla u\|_{\ell^2L^4([0,T_+)\times\mathbb{R}^2)}<\infty$, and also if a terminal one-window high-frequency tail $\mathcal{A}_{\mathrm{win}}(t)$ satisfies $\liminf_{t\uparrow T_+}\mathcal{A}_{\mathrm{win}}(t)<\varepsilon_*$. Thus any finite-time failure must force persistent high-frequency concentration on all sufficiently late critical parabolic windows.

This paper studies finite-terminal continuation and subthreshold global existence for the two-dimensional Landau–Lifshitz–Gilbert equation in the $H_Q^2$ strong-solution class. The proof shows that if a maximal strong solution terminates at a finite time and the full-frequency $L^4_{t,x}$ norm of $\nabla u$ is finite on a terminal time window, then the strong solution can be continued. Furthermore, the solution can still be continued as long as the ultraviolet tail above one fixed dyadic cutoff belongs to $L^4_{t,x}$. As applications, we obtain global strong solutions when $E(u_0)<4\pi$; in the degree-zero sector, this energy threshold can be raised to $8\pi$.

A bounded time-dependent drift together with an arbitrarily small real potential can produce double exponential decay of the full spatial $L^2$ norm. We prove matching quantitative two-time lower bounds for energy solutions of $\partial_tu-\Delta u=vu+w\cdot\nabla u$ on $\mathbb{R}^n$, for $n\geq3$, when $w\in L^\infty_{t,x}$ and either $v$ is sufficiently small in $L^\infty_tL^{n/2}_x$ or $v\in L^\infty_tL^p_x$ for some $n/2<p\leq\infty$, without a smallness assumption. In particular, a nonzero global solution satisfies a double exponential lower bound at large times. This decay scale is optimal for these coefficient classes: In every dimension $n\geq3$, we construct a real bounded drift, a real potential arbitrarily small in the critical space and in any prescribed subcritical $L^p$ space, and a nonzero real energy solution with double exponential decay. The construction localizes Rowan's torus example by absorbing the localization error.

We identify a common action–density structure for the Hartree and Vlasov–Riesz equations and their coupling. Along free rays, the feedback between nonlinear actions and rescaled densities governs wave phase modulations and kinetic momentum translations. This structure persists under the exchange of sources, providing convergent modified profiles and an explicit recursive construction of the asymptotic corrections. We thereby establish small-data global existence and modified scattering in three spatial dimensions for the Hartree equation, the Vlasov–Riesz equation, and the coupled Vlasov–Hartree system, with interaction potential $\lambda|x|^{-\alpha}$, $\lambda\in\mathbb R$ and $0<\alpha\leq 1/2$. In this strongly long-range regime, the asymptotic action contains finitely many successive corrections beyond its leading term. To our knowledge, these are the first forward modified-scattering results for all three models throughout this range.

In this paper, we investigate how spatial heterogeneity affects the stability of positive equilibria in one-dimensional predator–prey reaction–diffusion systems with Neumann boundary conditions. Our examples show that spatial heterogeneity can lead to instability even when the positive equilibrium is unique. For the smooth, spatially heterogeneous coefficient families constructed here, this equilibrium loses linear stability as a parameter in the reaction coefficients varies. At the critical parameter value, a simple conjugate pair of eigenvalues crosses the imaginary axis transversely, and a Hopf bifurcation produces a local branch of strictly positive, nonstationary classical time-periodic solutions. The examples include two families with spatially concentrated prey, one with an everywhere positive and the other with an everywhere negative predator intrinsic growth rate. A third family has a spatially concentrated predator population and an everywhere negative predator intrinsic growth rate.

We study the normal-mode stability of a flat interface between compressible Euler flow in the air and incompressible Euler flow in the water, including gravity and surface tension. The base state is quiescent water beneath a hydrostatic, constant-entropy shear flow $U(z)$ with $U(0)=0$, so the atmosphere is neutrally stratified. For a small ratio $\varepsilon$ of the air density at the interface to the water density, and for uniformly subsonic wave-relative flow, we establish a compressible analog of the critical-layer instability of Miles, extending the rigorous incompressible theory of Bühler, Shatah, Walsh, and Zeng. For a real phase speed $c_r$, the singular coefficient at a regular critical level $U(z_c)=c_r$ has a residue proportional to $Q=U''+gU'/a^2=\bar\rho_0(U'/\bar\rho_0)'$, with gravitational acceleration $g$, sound speed $a$, and normalized base-state air density $\bar\rho_0$. Thus the gradient of specific vorticity replaces the curvature $U''$ of the incompressible theory. We prove a limiting absorption principle at regular, non-resonant phase speeds and a residue formula for the imaginary part of the boundary value $Z_{\mathrm a}^+(k,c_r)$ of the air-side pressure response at wavenumber $k>0$. Let $c_k>0$ be the phase speed of the free gravity-capillary wave. If $\mathrm{Im}\,Z_{\mathrm a}^+(k,c_k)>0$, then, for sufficiently small $\varepsilon$, the dispersion relation has exactly one unstable root $c_\varepsilon$ in a neighborhood of $c_k$ that does not depend on $\varepsilon$. A sufficient condition is $Q\le0$ at every critical level and $Q<0$ at the highest one. As the Mach number $M\to0$, $Z_{\mathrm a}^+$ converges to its incompressible counterpart at rate $O(M^2)$. Under the corresponding incompressible sign condition, the unstable root satisfies $|c_\varepsilon(M)-c_\varepsilon(0)|\le C\varepsilon M^2$ with $C$ independent of $\varepsilon$ and $M$.

This paper is concerned with the effect of spatial heterogeneity on the linear stability of unique positive steady states for a nonlocal population equation. The main difficulty is that, because of the nonlocal term, the linearized operator need not be self-adjoint or satisfy the maximum principle, so the usual stability arguments based on the principal eigenvalue do not apply directly. We establish linear stability for sufficiently large diffusion and, under suitable conditions on the maximum set of the local growth ratio, for sufficiently small diffusion. In one space dimension, we also construct unstable positive steady states and obtain Hopf bifurcations for sufficiently small diffusion. Keeping the local coefficients fixed, we obtain either a supercritical or a subcritical Hopf bifurcation by changing the weight in the nonlocal integral. Our results show that uniqueness of a positive steady state does not imply its linear stability.

We consider the one-dimensional Vlasov–Poisson system with a repulsive harmonic potential. For repulsive self-interactions, we establish modified scattering for initial data of arbitrary size in some weighted Sobolev spaces without symmetry, compact support, or spatial moment assumptions. The proof uses a Hamilton-Jacobi momentum correction and the sign of the repulsive interaction to obtain large-data bounds. We also construct modified wave operators. These results complement the higher-dimensional theory in \cite{HK26,VR24,BVR25}. We expect our methods to be useful in the study of a broader class of kinetic equations with very long-range interactions.

These notes present the construction in \cite{euler} of finite-time blowup for the three-dimensional incompressible, unforced Euler equations on $\mathbb R^3$: \begin{equation} \partial_tu+(u\cdot\nabla)u+\nabla p=0,\qquad \nabla\cdot u=0. \end{equation} The construction successively adds oscillatory layers to smooth parent solutions. Each layer is constructed by a finite expansion and an exact correction, so that the resulting child solves Euler.

Electroconvective (EC) instability near ion-selective surfaces plays a key role in electro-chemical transport beyond the diffusion-limited regime. While EC instability in Newtonian electrolytes has been extensively investigated, the influence of viscoelasticity on the instability mechanism remains less understood. In the present study, we investigate the viscoelastic modulation of EC instability using linear stability, eigenspectrum, and mechanical-work analyses, supported by asymptotic pathway ordering. The analysis reveals a non-monotonic dependence on the Weissenberg number Wi, with the maximum growth rate and minimum critical voltage occurring near Wi ~ O(1). The mechanical-work balance separates electric, solvent-viscous, and polymeric contributions, and further decomposes the polymeric work into normal-stress-difference and shear-stress pathways according to their constitutive origins. For Wi << 1, the polymer stress recovers Newtonian viscous dissipation at leading order. As Wi approaches unity, the phase efficiencies of the Newtonian-like polymeric pathways decrease substantially, causing the negative polymeric work to collapse and the instability to intensify. At intermediate Wi, an additional minimum in the critical voltage appears at low wavenumbers. Eigenspectrum analysis attributes this feature to the selection of a slowly propagating mode that retains substantial negative polymeric work. At large Wi, this mode no longer determines the instability threshold, and the low-wavenumber minimum disappears. The normal-stress-difference and shear-stress work then nearly cancel, accompanied by stronger ESC-localized shear deformation and predominantly solvent-viscous dissipation. Thus, the non-monotonic response is governed by stress-deformation phase relations, solvent dissipation, and eigenmode selection rather than stress magnitude or a single polymeric contribution.

Consider a subcritical functional $Q$ associated with a $p$-Schrödinger operator defined on a domain $\Omega$ in noncompact Riemannian manifold. Employing the supersolution construction, we obtain a family of strictly positive critical Hardy weights for $Q$. The construction is based on the Green potential, with a strictly positive charge, which is assumed to vanish at infinity. An interesting new feature of this family, depending on a parameter $0\leq \varepsilon\leq 1$, is the transition from positive-criticality to null-criticality at $\varepsilon=0$. Moreover, we show that the best Hardy constant for this family is given by $c=[(p-1)/p]^{p-1}$, and that for $\varepsilon=0$ the corresponding Hardy weight is optimal. We furthermore, prove an analogous result on graphs.

This work establishes finite-time blow-up for the completely integrable defocusing Ablowitz-Ladik equation on both infinite and finite lattices. We first establish local well-posedness for arbitrary initial data in the space $\ell^\infty$ of uniformly bounded sequences, with no decay assumptions and under general boundary conditions. We then show that the unit circle acts as a separatrix for the lattice dynamics: sites with an initial modulus inside, on, or outside the unit circle maintain their modulus within their respective region for as long as the solution exists. As a consequence, initial data in the closure of the $\ell^\infty$ unit ball generate global-in-time solutions. On the other hand, the unit circle separatrix opens up the possibility for finite-time blow-up, which is restricted to sites with initial modulus outside the unit circle. We establish this scenario by constructing a class of initial data whose corresponding solutions are singular, with explicitly given exact blow-up times. We also construct solutions emanating from initial data with an arbitrarily large single-site excitation outside the unit circle that exist globally in time, a behavior driven by the convergence of the corresponding modulus to the unit circle equilibrium. In the presence of such solutions, along with other important classes of analytical, global-in-time solutions that remain outside the closed $\ell^\infty$ unit ball for all times, we examine the fundamental question of the dominant dynamics for general initial data outside the closed unit ball. By utilizing the constructed class of singular solutions as subsolutions for the general problem, we prove that the dynamics for the completely integrable defocusing Ablowitz-Ladik lattice is dominated by finite-time blow-up. We conclude with numerical studies of illustrative examples of general initial data

We establish lower bounds on the number of resonances for three-dimensional semiclassical Dirac operators with mass-type potentials. For a pair $\Dl_j=\Dl_0+\mbf{\beta}v_j$, $j=1,2$, satisfying the exterior analyticity, decay and non-degeneracy assumptions of the paper, we associate a relative distribution $\mu$ to the corresponding mass functions. We prove that if $E_0>1$ is a boundary point of $\supp\mu$, then every complex neighbourhood of $E_0$, and likewise of $-E_0$, contains in total at least $C\hbar^{-3}$ resonances of the two operators, counted with multiplicity. The proof transfers an analytic wave-front singularity of $\mu$ to the relative classical density of states and combines this with Khochman's local trace formula and a relative semiclassical trace formula. As a consequence, if $v$ is a single bounded mass-type potential, analytic with power decay in an exterior sector, with $\inf(1+v)>0$ and $\sup v>0$, then $\Dl=\Dl_0+\mbf{\beta}v$ has at least $C\hbar^{-3}$ resonances in every complex neighbourhood of $E_0=1+\sup v$ and of $-E_0$, for all sufficiently small $\hbar$. The exponent agrees with Khochman's upper bound, the two resonance families are related by spectral symmetry, and $E_0$ is necessarily a trapping energy. We also prove the relative semiclassical trace formula needed in the argument for general Hermitian matrix-valued perturbations.

The aim of the paper is to provide an extension of the averaging method to the periodic problem for quasilinear differential equations $(\phi(t,u'))' = \varepsilon f(t,u,u',\varepsilon)$, $u(0) = u(T)$, $u'(0) = u'(T)$, where $u$ and $u'$ take values in a Banach space $X$ and $\phi$ is a suitable homeomorphism between a neighborhood of $0$ in $X$ and $X$. The approach follows from a given extension of the averaging method to first order systems of the form $u' = g(t,v)$, $v' = h(t,u,v)$, $u(0) = u(T)$, $v(0) = v(T)$, in a product of Banach spaces, obtained from a direct application of the implicit function theorem in Banach spaces to an equivalent problem. We prove that every nondegenerate zero of the corresponding averaged equation generates, for sufficiently small nonzero values of the parameter, a unique nearby periodic solution. An application is given to an infinite system of coupled forced pendulums with relativistic acceleration.

We develop an abstract averaging method for equations $Mx=\varepsilon N(x,\varepsilon)$ in Banach spaces. The method applies when the equation can be separated into an averaged compatibility condition and a nonlinear complementary problem, and it allows the principal operator $M$ to be genuinely nonlinear. We prove that every nondegenerate zero of the corresponding averaged equation generates, for all sufficiently small nonzero values of $\varepsilon$, a nearby solution. Under a suitable small Lipschitz condition this solution is locally unique, while in the differentiable case it belongs to a unique local $C^1$ branch. The framework contains the abstract linear Fredholm averaging theory as a special case, but also covers quasilinear problems for which the principal operator has no invertible linearization at the reference state. We also determine the leading asymptotic profile of the solutions. Applications are given to a vector $p$-Laplacian Duffing system and to a nonsmooth problem in a Hilbert space.

We establish a natural generalization of the Hörmander condition for the multiplication of Besov distributions using the global wavefront set framework of Coriasco, Johansson, and Toft. Using this condition, we relate the microlocal perspective on multiplication of distributions to the classical paraproduct decomposition of pointwise multiplication arising in the paradifferential calculus of Bony.

We consider a system of elliptic equations, depending on a small parameter $\epsilon > 0$, which models the long range segregation of populations. The system has been previously studied in \cite{CL2} for the regularity of the free boundary in dimension $2$ and in \cite{ChPaTo26_2} and \cite{ChPaTo26_3} for the partial regularity of the free boundary in higher dimensions. In this paper, we consider the special case with $K = 2$ populations in an annular domain in arbitrary dimensions. Using the uniqueness and the rotational invariance of the solution, we show that the free boundary consist of concentric spheres. Moreover, by an application of the free boundary condition derived in \cite{CL2}, we show that the free boundary is uniquely determined. We also examine how the radius of the free boundary change according to the domain, the interaction distance and the boundary data. In particular, we show that the free boundary converges to that of the adjacent model as the interaction distance tends to zero. We also study the one parameter family of the elliptic system, in which the domain, the interaction distance, and the boundary data depends on $t$. We derive an ODE for the radius of the free boundary in dimension $2$ and dimension $n \geq 3$ separately. Several examples are given and discussed.

We study resonance expansions and localised long-time dynamics for three-dimensional semiclassical Dirac operators. For smooth Hermitian matrix-valued perturbations which are analytic outside a compact set, we first obtain exponential cut-off resolvent bounds away from the resonance set and polynomial bounds in thin resonance-free rectangles adjacent to the real axis. Combining these estimates with a local $\Oc(h^{-3})$ upper bound on the number of resonances, a pigeonhole selection of admissible contours, almost-analytic functional calculus, and a Cauchy–Green deformation, we derive resonance expansions for spectrally localised propagators. Positive-energy resonances contribute for large positive times, whereas negative-energy resonances contribute for large negative times. We then specialise to mass-type perturbations $\mbf{\beta}V(x)$ in a nontrapping energy window. Using the logarithmic resonance-free region established in our earlier work together with a quantitative FBI-transform/escape-function argument, we obtain a polynomial bound for the continued cut-off resolvent and deduce rapid local-energy decay. We subsequently refine this polynomial estimate by a finite-time propagation argument in the undistorted region. Introducing the maximal positive-sheet connection time $T_\chi$ associated with the spatial cutoff, we prove a geometric continued-resolvent estimate and, for every fixed $M$ below the resonance-free depth constant, derive the explicit local-energy decay threshold $T_N>T_\chi+(N+1)/M$. The same analysis gives a rigorous microlocal obstruction to a uniform $[h\log(1/h)]^{-1}$ bound for the full distorted resolvent and identifies a finite-flight-time amplification mechanism for the sandwiched continued resolvent at logarithmic depth.

We give sufficient conditions ensuring that, at every fixed positive time, the scalar backward component of a Markovian forward-backward stochastic differential equation with multidimensional forward process admits a density with respect to Lebesgue measure. Existing density criteria for BSDEs often obtain Malliavin non-degeneracy through sign or monotonicity assumptions. We develop a different route for a scalar backward component with an arbitrary-dimensional forward state. Under regularity assumptions and a structural compatibility condition on the generator, the terminal condition is only required to be non-constant. The key idea is to deduce Malliavin non-degeneracy from deterministic rigidity of the critical set of the decoupling field. We combine Malliavin calculus with unique continuation and backward uniqueness for the associated semilinear parabolic equation. Unique continuation precludes the spatial gradient of the decoupling field from vanishing on a set of positive measure unless it vanishes identically on that time slice, while backward uniqueness propagates such vanishing to the terminal time. Along the way, we establish a unique continuation property from sets of positive measure and a backward uniqueness result on the whole space for the linear parabolic systems arising from differentiated semilinear equations.

We study quadratically regularized optimal transport with quadratic cost in the regime of small regularization $\varepsilon$, where the support of the optimal coupling is sparse. For smooth marginal densities in $\mathbb{R}^d$, we show that the support contains the graph of the Brenier map. After centering by the Brenier image and rescaling by $\ell=\varepsilon^{1/(d+2)}$, the sections of the support have explicit ellipsoidal limits for $\varepsilon\to0$, except at boundary points, where the limits are half-space profiles. The optimal dual potentials admit expansions at order $\ell^2$, uniformly up to the boundary. We identify the leading coefficients, which consist of a common local profile and opposite global corrections determined by a linear Neumann problem. Finally, we analyze two approximations to the Brenier map, namely the gradient of the dual potential and the conditional mean of the coupling. We obtain the sharp $L^p$ rate $\ell^{1+1/p}$ with exact leading constants and further identify the leading interior and boundary biases. Taken together, our results illustrate that quadratic regularization induces an accurate sparse approximation of classical optimal transport.

We establish interior and global Hölder estimates for linearized Monge-Ampère equations in divergence form in all dimensions $n\geq 3$, when the Hessian determinant of the convex Monge-Ampère potential is bounded above and below by positive constants and the vector field on the right-hand side is bounded. The estimates use the $L^p$ norm of the solution for any $p>1$. A key ingredient is an $L^1$ estimate, with appropriate decay rates, for the gradient of the Green's function of the linearized Monge-Ampère operator in small sections, uniform with respect to the pole. As applications, we obtain uniform Hölder estimates for the time derivatives of the primal and dual potentials in the three-dimensional periodic dual semigeostrophic system when the initial density is bounded away from zero and infinity, and establish a linear convergence rate for periodic homogenization of the Monge-Ampère equation.