Papers

We study a jointly exercised American condor contract whose payoff is a call condor payoff plus a positive cash amount. Its plateau and positive tails produce two inner exercise boundaries and two outer cash-exercise boundaries. We construct the unique bounded solution of the associated variational inequality, strong on each side of the plateau, and identify it with the optimal stopping value. We prove monotonicity and continuity of the four boundaries and determine their limits at expiry. Following Broadie and Detemple, we express the inner boundaries as cutoffs of the boundaries of uncapped contracts with the same cash components. We also prove call-side convexity and volatility monotonicity under a sufficient condition, solve the stationary problems explicitly and present numerical results.

Let $\lambda_k$ be the $k$th eigenvalue of the Dirichlet Laplacian on an open set $\Omega\subset\mathbb R^n$ of finite positive measure. The direct individual consequence of the Berezin–Li–Yau sum inequality has leading coefficient $n/(n+2)$ relative to the Weyl term. The main contribution of this paper is a strict improvement of this coefficient. Energy orthogonality gives a frequency-dependent cap on the Fourier density of the first $k$ eigenfunctions. Combining this cap with the standard Bessel bound and a bathtub principle for the radial capacity yields \[ \lambda_k\geq c_n(2\pi)^2\omega_n^{-2/n} |\Omega|^{-2/n}k^{2/n}, \qquad \frac{n}{n+2}<c_n<1, \] for every $k\geq1$ and every $n\geq2$, without boundary regularity. The constants are characterized by explicit scalar equations; in dimension two, $c_2=0.5383068077\ldots$, giving a $7.66\%$ improvement over the individual Li–Yau coefficient. We emphasize that this improves the leading coefficient in the individual eigenvalue bound and the new constant $c_n$ is independent of the geometry and index $k$.

The Vlasov-Maxwell system with Newtonian particle transport is linearised about the Poisson equilibrium on $\mathbb{R}^3$, with the speed of light $c$ treated as a large parameter. For uniformly controlled initial data, with transverse fields that may remain of order one, the difference between the Vlasov-Maxwell perturbation and its linearised Vlasov-Poisson counterpart is bounded uniformly in time by their initial discrepancy plus a term of order $c^{-1}$. The same estimate holds for their scattering states, and the rate in $c$ is sharp. The longitudinal dynamics are governed by the same Volterra equation as in linearised Vlasov-Poisson and undergo Landau damping. The transverse fields exhibit Klein-Gordon-type dispersion, and control of their accumulated effect along free transport yields the uniform comparison.

We prove that a smooth, regular, asymptotically flat stationary vacuum black hole has a nonextremal Kerr exterior if no null geodesic orthogonal to its stationary Killing field is trapped, establishing the rigidity conjecture of Alexakis, Ionescu, and Klainerman. In the rotating case, we start with the rotational Killing field constructed near the bifurcation sphere and study a maximal domain carrying its complete circle action. The main difficulty is to recover boundary regularity and the curvature condition needed for local Killing extension. We derive uniform estimates along complete rotational orbits from nontrapping and combine them with convexity along short null geodesics. These estimates show that a nonempty remaining boundary in the ergoregion is a timelike photon surface for the stationary quotient metric. Its null geodesics lift to spacetime null geodesics orthogonal to the stationary field. The vacuum circularity identities give their continuation through contact with the ergosurface, keeping their stationary projections in a compact set. This contradicts nontrapping and yields the global rotational symmetry required by black hole uniqueness.

We classify all smooth local conservation laws of three-dimensional homogeneous incompressible Euler flow in velocity and pressure, allowing explicit time and position and derivatives of arbitrary finite order. Modulo currents vanishing on solutions and currents with identically zero divergence, the classical mass, momentum, angular-momentum, kinetic-energy and helicity balances, together with generalised time-dependent momentum, exhaust the classification. We also classify all smooth finite-order local symmetry characteristics and adjoint symmetries: differential expressions satisfying the linearised Euler equations and their formal adjoint on solutions, respectively. Every symmetry characteristic is equivalent on solutions to one induced by a point transformation. Every adjoint symmetry has a multiplier representative, whose product with the Euler equations is identically a divergence. These representatives exhaust the restrictions of all local conservation-law multipliers to solutions. The conservation-law proof has three stages: classification of adjoint symmetries using their determining equations, verification of explicit multiplier identities, and the correspondence between multiplier restrictions and current classes. A common analysis of highest-derivative responses reduces both symmetry problems to order at most one.

We prove uniqueness, up to phase rotations and translations, of the ground state for the $L^2$-critical boson star equation \[ (\sqrt{-\Delta}+1)Q =\bigl(|x|^{-1}* |Q|^2\bigr)Q, \qquad x\in\mathbb R^3. \] We also determine the kernels of the real and imaginary linearized operators. As a consequence, we show the Lieb-Yau uniqueness conjecture holds when the mass is sufficiently close to the Chandrasekhar mass.

We consider the two-dimensional barotropic compressible nematic liquid crystal flows with a planar director in the whole space or in the half-space. Under the assumptions that the shear viscosity is a positive constant and that the bulk viscosity is given by $\lambda(\rho)=\rho^\beta$ with $\beta>4/3$, we establish the global existence and uniqueness of strong solutions. It should be mentioned that our results are obtained without any restrictions on the size of the initial data and allow for the presence of vacuum. In particular, the initial orientation field is not required to satisfy any geometric angle condition.

In this paper, we investigate two-dimensional compressible nematic liquid crystal flows with a planar director. We assume that the shear viscosity is a positive constant and that the bulk viscosity is given by $\lambda(\rho)=\rho^\beta$ with $\beta>4/3$. In the periodic domain or in a bounded simply connected smooth domain with Navier-slip and Neumann boundary conditions, we establish the global existence and uniqueness of strong solutions without any restrictions on the size of the initial data. In particular, no geometric angle condition is imposed on the initial orientation field. Moreover, we derive a time-uniform upper bound for the density and establish exponential decay estimates for the strong solutions.

This paper studies the two-dimensional nonhomogeneous incompressible nematic liquid crystal flows with planar orientation fields taking values in $\mathbb{S}^1$. For the initial-boundary value problem in bounded domains with density-dependent viscosity, we establish the global existence and exponential decay of strong solutions with initial density allowing vacuum, provided that $\|\nabla \mu(\rho_0)\|_{L^q}$ is sufficiently small for some $q>2$. The smallness condition is automatically satisfied when $\mu$ is constant, and hence the result yields the global existence of strong solutions for arbitrarily large initial data in the constant viscosity case. Furthermore, for the Cauchy problem with constant viscosity and either vacuum or non-vacuum far-field density, we establish the global existence and large-time decay rates of strong solutions for arbitrarily large initial data. These results are obtained without imposing any geometric angle conditions on the initial orientation field.

We study the full angular dependence of the linearized Schrödinger form of the Calderón problem on the unit disk. We obtain an explicit continuous spectral diagonalization of its normal operator and hence a generalized singular system for the linearized forward map. To our knowledge, this is the first such diagonalization for the full angularly dependent, multiplicity-normalized linearized measurement map on the disk.

In this contribution, a fluid flow problem on a general deforming domain for a Newtonian fluid in two and three space dimensions with artificial compressibility approximation is studied. We prove an estimate on the integral equicontinuity in time of the weak solutions under suitable domain regularity assumptions, which is independent on the compressibility parameter and serves as an alternative compactness argument for the convergence of weak solution sequences for vanishing compressibility. The corresponding estimate is obtained by remapping the problem onto a fixed reference domain and using appropriate divergence-preserving testfunctions involving the difference of two solutions at different points in time, thus defined with respect to different domains/coordinates.

We establish global well-posedness and a small-noise large deviation principle for the two-dimensional Dean-Kawasaki equation with Coulomb interaction $\partial_t\rho^\varepsilon=\Delta\rho^\varepsilon-\nabla\cdot(\rho^\varepsilon(V*\rho^\varepsilon))-\sqrt{\varepsilon}\nabla\cdot(\sqrt{\rho^\varepsilon}\circ\xi^{K(\varepsilon)})$, where $V=\lambda_V(\cos(\alpha_V)\nabla G+\sin(\alpha_V)J\nabla G)$. Here $\lambda_V\geq0$ is the interaction strength, $\alpha_V$ is the interaction angle, $G$ is the mean-zero Green function of $-\Delta$ on $\mathbb{T}^2$, $J$ is rotation by $\pi/2$, and $\xi^K$ is a finite-mode approximation of space-time white noise. For finite-entropy initial data of mass $M$ satisfying $\lambda_V\max\{\cos(\alpha_V),0\}M<8\pi$, the equation with the exact square-root coefficient is globally pathwise well posed at every finite Fourier cutoff in the class of stochastic kinetic solutions. Following Fehrman and Gess (arXiv:1910.11860), we prove a large deviation principle on $L^1((0,T)\times\mathbb{T}^2)$ under the scaling $K(\varepsilon)\to\infty$ and $\varepsilon K(\varepsilon)^4\to0$, with good rate function given by the quadratic control problem for the skeleton equation.

We establish quantitative propagation of chaos in total variation for the unregularized three-dimensional repulsive Vlasov–Poisson–Fokker–Planck (VPFP) particle system. For tensorized initial data satisfying weighted Sobolev regularity and a Gaussian moment, we prove that for every $0<b<1/3$ there exist $\tau_b>0$ and $C_b\ge1$, independent of $N$ and $k$, such that \[ \sup_{0\le t\le\tau_b}\|F_{N,k}(t)-f_t^{\otimes k}\|_{\mathrm{TV}} \le C_b^kN^{-b}, \qquad N\ge1,\quad 1\le k\le N. \] More generally, for repulsive Riesz potentials with singularity $|x|^{-s}$ on $\mathbb{T}^d$ and $\mathbb{R}^d$, $0<s<d/2$, we obtain $C^k\bigl(N^{-1/q}+N^{-(d/s-2)}\bigr)$ for $q\ge2$ and $q>d/(d-s-1)$. The proof combines conditioning of the product initial law, auxiliary $k$-particle Fokker–Planck equations, and weighted $L^q$ estimates for the BBGKY hierarchy. The level-$(k+1)$ interaction term is controlled by a weighted Hölder estimate under the local condition $K\in L^{q'}_{\mathrm{loc}}$; for $d=3$ and $s=1$, this condition is $q>3$. On $\mathbb{R}^d$, polynomial spatial weights control the far field. For each fixed particle number, the singular particle dynamics are globally well posed and have no collisions. This permits the force regularization to be removed.

We prove finite time blow-up from smooth, radial, compactly supported initial data for the energy supercritical defocusing nonlinear Schrödinger equation $i\partial_tu+\Delta u-|u|^{p-1}u=0$ in $\mathbb R^d$, for $(d,p)\in\{(3,27),(4,9),(5,7)\}$. The solutions are asymptotically self-similar, their $L^\infty$ norm grows at the scaling rate $(T-t)^{-1/(p-1)}$, and their critical Sobolev norm $\dot H^{s_c}$ diverges. This closes the conjecture of Bourgain \cite{Bourgain2000} on global well-posedness and scattering for the defocusing energy supercritical equation in dimensions three and four. The proof relies on a shooting argument and is computer-assisted.

The inverse scattering problem for the one-dimensional nonlinear Dirac equation \begin{align*} \begin{cases} i(u_t+u_x)+v=\mathcal{N}_1(u,v),\\ i(v_t-v_x)+u=\mathcal{N}_2(u,v) \end{cases} \end{align*} is studied. We assume that the unknown nonlinearities $\mathcal{N}_j(u,v)$ ($j=1,2$) of the equation belong to $C^\infty(\mathbb{C}^2;\mathbb{C})$ and satisfy $(\partial_\mathbf{z}^\alpha \mathcal{N}_j)(\mathbf{z})=O(|\mathbf{z}|^{\max\{ 5-|\alpha|,0 \}})$ ($|\mathbf{z}|\to 0$) for any multi-index $\alpha \in (\mathbb{N} \cup \{ 0 \})^4$. Here, $\mathbf{z}=(z_1,z_2)\in \mathbb{C}^2$, $ \partial_\mathbf{z}^\alpha= (\partial_{z_1},\partial_{\overline{z_1}},\partial_{z_2},\partial_{\overline{z_2}})^\alpha, $ and $\partial_{z_1},\partial_{\overline{z_1}},\partial_{z_2},\partial_{\overline{z_2}}$ are the Wirtinger differential operators. Under some additional assumptions on $\mathcal{N}_j$, we establish a reconstruction formula for $(\partial_\mathbf{z}^\alpha \mathcal{N}_j)(0)$ ($|\alpha|\ge 5$) using the knowledge of the scattering operator for the equation. Although such results were established for the nonlinear Schrödinger equation [Sasaki 2024] and the nonlinear Klein-Gordon equation [Sasaki 2025], the methods in [Sasaki 2024, 2025] are not directly applicable to the nonlinear Dirac equation due to its system nature. Specifically, it is quite difficult to prove the invertibility of a matrix associated with the input data. To overcome this difficulty, we introduce a new method based on the massless limit for the free solutions.

We establish the complete integrability of the defocusing Calogero–Sutherland derivative nonlinear Schrödinger equation \begin{equation}\label{CS-abstract}\tag{CS} i\partial_t u +\partial_x^2 u + 2i u \partial_x\,\Pi\!\left(|u|^2\right)=0,\qquad x\in\mathbb T, \end{equation} by constructing a nonlinear Fourier transform \begin{equation*} \Phi : u\in L^2_+(\mathbb T)\mapsto\ (\zeta_n(u))_{n\geq 0}\in \ell^2(\mathbb N_0),\qquad \zeta_n(u)=\sqrt{\gamma_n(u)}\;e^{\,i\arg\langle u\mid f_n\rangle}, \end{equation*} built from the eigenvalues $(\lambda_n)$, the spectral gaps $\gamma_n:=\lambda_n-\lambda_{n-1}-1\geq 0$ and the eigenfunctions $(f_n)$ of the Lax operator $L_u$. We prove that $\Phi$ is a norm-preserving homeomorphism. In these coordinates, the flow becomes the explicit rotation $\zeta_n(t)=e^{-i\omega_n t}\zeta_n(0)$ for all $n$, whose frequencies $\omega_n$ depend only on the conserved spectrum $(\lambda_n)$. As consequences, every orbit of \eqref{CS-abstract} is precompact in $L^2_+(\mathbb T)$, and $t\mapsto u(t)$ is Bohr almost periodic. For finite-gap data, we obtain quasi-periodicity in time and a sharp criterion for genuine periodicity.

We consider a Whitham-Boussinesq type system with surface tension arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water, which is characterized by a nonlinearity parameter $0<\epsilon\le 1$, a shallow water parameter $0<\mu\le 1$ and a nonnegative surface tension parameter $\beta$. In this paper, we establish well-posedness of the associated Cauchy problem for $\beta>1/3$ on a time scale of order $ ( \beta_\ast^{ 1/4} \mu^{ 1/4} h_0 \epsilon^{-1} )^{4/3}$ in one dimension and of order $( \beta_\ast^{ 1/4} \mu^{ 1/4} h_0 \epsilon^{-1})^{2-}$ in two dimensions, where $\beta_\ast= \min(|3\beta-1|, 1/2)$, assuming a non-cavitation condition with a parameter $h_0>0$. In addition, we established well-posedness for the classical Whitham equation with either $\beta=0$ or $\beta>1/3$ on the timescale of order $ ( \beta_\ast^{ 1/4} \mu^{ 1/4} \epsilon^{-1} )^{4/3}$. These results explicitly capture how the existence time depends on all the parameters involved. Our proofs rely on dispersive and Strichartz estimates combined with energy estimates.

In this paper, we study the Dirichlet eigenvalue problem for the logarithmic Hardy operator, defined as the logarithmic Laplacian with a critical logarithmic potential, in a bounded Lipschitz domain containing the origin. We first establish a logarithmic Hardy inequality and use it to identify the associated energy space, showing that the embedding into \(L^2\) is compact when the perturbation parameter exceeds the endpoint value \(-1\), but fails to be compact at that endpoint. For parameters above \(-1\), we develop a complete variational spectral theory, prove that the problem admits a discrete sequence of eigenvalues tending to infinity, characterized by successive minimizations over orthogonal complements, whose eigenfunctions form a complete orthonormal basis of \(L^2\). We further establish a uniform lower bound for the first eigenvalue, a scaling property under domain dilation that determines a critical radius for the sign of the first eigenvalue and leaves the eigenvalue gaps invariant.

This paper investigates the $L_p$-dual mixed Euclidean-Gauss Minkowski problem associated with the $L_p$-dual mixed Euclidean-Gaussian curvature measure. The solvability of this problem is equivalent to that of a class of Monge-Ampère type equations on the unit sphere. Under smooth assumptions, We establish the existence of solutions for $p<1$, $q<0$, and $q<p$ and uniqueness of solutions for $p\geq q$ to the aforementioned Monge-Ampère type equations, respectively. Furthermore, we obtain a complete solution concerning the existence part of this problem when $p<1$ and $q<0$.

We study the following curvature equation with two singular sources on a flat torus $$ \Delta u+e^u=8\pi(\delta_p+\delta_{-p})\quad\text{on } E_\tau:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau), $$ where $\delta_p$ denotes the Dirac measure at $p$. The case $\wp'(2p)=0$ was solved by Kuo (J. Differential Geom. 2026). In this paper, we study the general case $\wp'(2p)\neq 0$. A novel phenomenon for $\wp'(2p)\neq 0$ is that the Hill discriminant of the associated generalized Lamé equation has an essential singularity. We develop new ideas to determine the intersection of two conditional stability sets completely, which is quite surprising since it seems impossible to determine the exact structure of these two sets due to the essential singularity. This leads to a sharp criteria for the existence and non-existence of solutions when $E_\tau$ is a rectangular torus and $\wp(p)\in \mathbb{R}$.

We prove weak existence and uniqueness in law of $[0,1]$-valued solutions for a class of scalar stochastic heat equations with measurable drifts $b : [0,1] \to \mathbb{R}$, driven by Wright–Fisher noise. The only assumptions we require on the drift are Borel-measurability, and the one-way inequalities $-Cu\leq b(u)\leq C(1-u)$ for some $C<\infty$, compatible with the necessary condition $b(1)\leq 0\leq b(0)$. These are vastly more general conditions than were previously available. Our proof relies on a stochastic duality between the solution of the stochastic heat equation, and a voting scheme running along the graph of a branching-coalescing particle system, generalising the class of drifts for which a dual process is available. Our results and their proof put the regularisation-by-noise result of Barnes, Mytnik and Sun, and its explanation, on a much more general footing.

In this paper, we study a finite-data inverse nodal optimization problem for radial Schrödinger operators in an arbitrary fixed angular-momentum sector. The analysis is built directly at the Friedrichs endpoint and in the physical weighted space $L_d^p$, so that the singular radial geometry is retained rather than replaced by a regular one-dimensional model. The main purpose of this paper is to provide a singular Friedrichs finite-data variational framework valid in every angular-momentum sector, thereby extending the existing finite-data variational theories concerning either regular one-dimensional operators or the radial sector $\ell=0$. By means of a Volterra representation of the Friedrichs branch, we prove weak continuity and continuous Fréchet differentiability of nodal radii, exact realization of compatible same-mode nodal data, existence of optimal potentials, and finite-codimensional constraint geometry. The same framework also incorporates mixed angular-momentum and spectral–nodal observations through finite-dimensional transversality. Remarkably, a global uniqueness theorem is established for inward displacements of the unique interior node of the second mode in the radial sector \(\ell=0\). For a constant reference potential and \(p>(d+2)/2\), every such displacement admits a unique global optimizer. Unlike local inverse-mapping or one-dimensional integrability arguments, the proof first selects the admissible critical sign globally and then reduces every minimizer to a scalar mass-balance equation between a focusing ball branch and a logistic annulus branch. The strict opposite monotonicity of the two weighted masses makes the balance parameter unique, providing a global rigidity mechanism over the entire inward-displacement regime.

We introduce various classical, foundational results on the regularity theory of harmonic maps, with focuses on the energy minimising harmonic maps. Topics include inner and outer variations, the monotonicity identity, the $\varepsilon$-regularity theorem, blow-ups and tangent maps, and the dimension estimate and stratification of singular sets. We also discuss the analogous theory for stationary harmonic maps and mention some significant recent developments.

We study the nonlinear evolution equation $$ \partial_t u+(G+G^\delta)u=I_\alpha(|u|^\rho) \quad\text{on }\mathbb{R}^{N+k}, $$ where $G$ is the nonnegative self adjoint realization of Grushin operator, $0<\delta<1$, and $I_\alpha$ is the potential operator with kernel $|z|^{-\alpha}$, $0<\alpha<N+k$. We develop a well-posedness theory for initial data in Marcinkiewicz spaces, allowing singular profiles outside the corresponding Lebesgue spaces. Using spectral calculus, subordination, and interpolation, we establish Lebesgue and Lorentz smoothing estimates for the mixed semigroup $e^{-t(G+G^\delta)}$, which distinguish the second-order behaviour at short times from the fractional decay at large times. In the subcritical regime, considering $$ \frac1\beta=\frac2Q+1-\frac{\alpha}{d}, \qquad \frac1{\beta_\delta}=\frac{2\delta}{Q}+1-\frac{\alpha}{d}, $$ where $Q$ is homogenous dimension, we prove local existence, uniqueness, Lipschitz dependence on the initial data, and a blow-up alternative in $L^{p,\infty}$ when $$ \rho<1+\frac{p}{\beta}. $$ At the critical relation $\rho=1+p/\beta$, a Yamazaki-type estimate and Lorentz duality yield global mild solutions for sufficiently small data in $L^{p,\infty}$. In the supercritical regime $\rho>1+p/\beta$, well-posedness is recovered in higher-integrability Marcinkiewicz spaces $L^{q,\infty}$ satisfying $$ \beta(\rho-1)<q<\beta_\delta(\rho-1), $$ with global existence for sufficiently small initial data. In all cases, the solutions attain their initial data in the weak-$*$ sense. These results extend the Marcinkiewicz-space theory to mixed Grushin diffusion with a spatially nonlocal potential source and reveal the role of the two competing diffusion scales in determining the admissible integrability regimes.

We identify a directional tangency mechanism responsible for anomalous far-field decay in Schrödinger equations. For constant-coefficient operators, we introduce a directional decomposition of the limiting resolvent based on the geometry of the Fermi surface and the observation direction. In two dimensions, we show that an analytic source produces a super-algebraically small tangency contribution, whereas a transverse non-analyticity located on the directional tangency set generates an algebraic contribution of order \(R^{-(2\alpha+1)}\). Shifting the singularity away from the tangency set restores super-algebraic decay, showing that geometric alignment is essential to the anomalous behavior. We further construct a bounded, decaying real-valued potential for which the same mechanism occurs in the first Born correction through the effective source \(Vu_0\). In a suitable long-range regime, the resulting tangency contribution decays more slowly than the conventional two-dimensional far-field term.

We study the nonlinear stability of a composite wave consisting of a traveling wave and a rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis. We prove that if the initial value is a small $H^1$-type perturbation of composite wave, then the system admits a global solution that converges toward the composite wave with an absolutely continuous shift. The proof combines the weighted relative-entropy mechanism for viscous shocks with the energy structure of rarefaction waves. A key ingredient is the inclusion of a rarefaction modulation factor in the weighted relative entropy; its derivatives combine with the terms generated by the rarefaction profile to produce a rarefaction dissipation. Moreover, the spatial separation of the two waves yields time-integrable interaction errors caused by the non-exact superposition. Since diffusion acts only on the density component, the full $H^1$ estimate is closed by exploiting the coupling structure of the system to recover the missing dissipation of the hyperbolic component.

Let $N\geq 3$, $2<q<p<2^*$, and let $V\in L^\infty(\mathbb{R}^N)$ be real-valued and $\mathbb{Z}^N$-periodic. We assume that $0$ belongs to a finite spectral gap of the periodic Schrödinger operator $L=-\Delta+V$. We prove that there exists $\lambda_\infty>0$ such that, for every $\lambda\geq\lambda_\infty$, the competing-power equation $$ -\Delta u+V(x)u=|u|^{p-2}u-\lambda |u|^{q-2}u \quad\text{in }\mathbb{R}^N $$ has a nontrivial solution $u_\lambda\in H^1(\mathbb{R}^N)$. Moreover, $$ \|u_\lambda\|_{H^1(\mathbb{R}^N)}+\|u_\lambda\|_{L^\infty(\mathbb{R}^N)} \leq C\lambda^{-1/(q-2)}. $$ We also show that, after the natural rescaling and lattice translations, a sequence of such solutions converges weakly to a nontrivial solution of the pure defocusing equation $Lv=-|v|^{q-2}v$.

This paper concerns continuous sonic-supersonic flows in a two-dimensional infinitely long divergent nozzle with straight solid walls. For a given inlet that is a small perturbation of an arc centered at the vertex of the nozzle, we establish the global existence and uniqueness of a continuous sonic-supersonic potential flow which is close to the radially symmetric sonic-supersonic flow. The flow is sonic at the inlet, with its velocity along the normal direction, and becomes supersonic in the nozzle. Such a sonic-supersonic model is governed by a quasilinear nonstrictly hyperbolic equation with degeneracy at the inlet, and the degeneracy is weak in the sense that there exist exactly two distinct characteristics from each sonic point, except for those at the walls, pointing into the supersonic region. One of the crucial ingredients of the analysis is to obtain the precise asymptotic behavior near the sonic inlet by the method of characteristics, which is essential for establishing the local existence of continuous sonic-supersonic flows. Another one is to extend the local solution globally and derive uniform estimates in the supersonic region.

Let $M_1$, $\cdots$, $M_\ell$ be complete manifolds of the same dimension, where $2\le \ell\in\mathbb{N}$. Suppose that each $M_i$ satisfies two side Gaussian bounds. If some of these manifolds are parabolic and there exists a constant $1\le n_i\le 2$ such that for some $x_i\in M_i$ with $1 \le r \le R < \infty$, $$c_i\left(\frac{R}{r}\right)^{n_i}\le \frac{Vol_{M_i}(B(x_i,R))}{Vol_{M_i}(B(x_i,r))}\le C_i\left(\frac{R}{r}\right)^{n_i},$$ by assuming that all the manifolds satisfy the relative connectedness of the annuli ($RCA$) condition introduced by Grigor'yan and Saloff-Coste, we show that the Riesz transform $\nabla \mathcal L^{-1/2}$ is bounded on $L^q(M)$ for each $1<q<2$ on the gluing manifold $M=M_1 \# M_2 \# \cdots \# M_\ell$.

We are concerned with the following partial differential equations arising from urban crime modeling: \begin{equation} \label{main} \begin{cases} u_t = \Delta u - \chi \nabla \cdot \left( u \dfrac{\nabla v}{v} \right) - uv + B_1 + ru - \mu u^2, \\[4pt] v_t = \Delta v - v + uv + B_2, \end{cases} \end{equation} under no-flux boundary conditions in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$ with $n \geq 2$, where $\chi$, $r$, and $\mu$ are positive constants. It is shown in this paper that if the nonnegative source terms $B_1$ and $B_2$ are sufficiently regular and \begin{equation*} \mu > \frac{3}{n} + \frac{1}{n}\left( \chi n + \frac{n(\chi(n-1)-2)^2}{4(n-1)} \right)^{\frac{n+1}{n}} \cdot \left( \frac{(2n+\sqrt{n})^2}{2n-1} \right)^{\frac{1}{n}} + n^{12n+3}, \end{equation*} then the system \eqref{main} with suitably regular initial data possesses a global classical solution. Moreover, under the assumption that \begin{equation*} \label{cond.B2} \inf_{t>0} \int_\Omega B_2(\cdot,t) > 0, \end{equation*} the solutions are uniformly bounded in time.