Papers

This work presents analytical tools for studying functions defined on spaces of measures, together with two applications: mean field game master equations and first order Hamilton-Jacobi-Bellman equations on the set of probability measures. The first part introduces several notions of differentiability for such functions - vertical (or flat) and horizontal derivatives - as well as associated notions of convexity, monotonicity and sub-differentiability. The second part is concerned with master equations, for which uniqueness and stability of monotone solutions are established, and existence is discussed. The third part studies mean field optimal control problems and characterizes their value functions as the unique viscosity solutions of the associated Hamilton-Jacobi-Bellman equations. This work is intended to be accessible to readers unfamiliar with spaces of measures.

In this paper we analyze the Kuramoto-Sivashinsky equation (KSE), a model of flame-front propagation, on a two-dimensional square torus of arbitrary size $2 L$ with $L > \pi$. In this case, the linearized equation at the origin admits a finite number of growing modes, which corresponds to the positive eigenvalues of the linear operator $- \Delta^2 - \Delta$. The problem of analyzing the long-time behavior of solutions of the 2D KSE in two spatial dimensions remains largely open; the only global existence results are for sufficiently small tori, or for sufficiently anisotropic and thin domains, due to the lack of good a priori estimates. The main purpose of this paper is to analyze the instability around growing modes at the nonlinear level. More precisely, we consider the maximal growing mode $\lambda_0$ and we show that there is a finite dimensional manifold of initial data of size $\varepsilon$ arbitrarily small such that the corresponding solutions become of size $O(1)$ over a time-scale of order ${\rm log}(\varepsilon^{- 1})$. The proof is based on several ingredients such as a sharp quantitative construction of an approximate solution bifurcating from the maximal linearly growing mode, a fixed point argument with exponential weights to construct local in time solutions, and a continuation argument based on sharp energy estimates and para-differential calculus.

We study the Liouville equation $-\Delta u = e^u$ on a complete, connected, non-compact, boundaryless Riemannian surface $(M, g)$ with non-negative Ricci curvature. Assuming only some asymptotic lower bound on the solution, we establish classification results for both the solutions and the ambient manifold, discussing also their optimality. Our results reveal a close connection between the volume growth of the manifold and the classification of both the solutions and the underlying manifold.

We study the simultaneous recovery of the initial source and sound speed for the scalar wave equation from a single boundary measurement. Although the recovery of either parameter separately is well understood under suitable geometric hypotheses, simultaneous recovery remains open in general, and stable recovery is further obstructed by the inherent instability of the linearized problem. We show that both uniqueness and stability can be obtained when the two unknowns are coupled through a prescribed constitutive relation arising from a common underlying material. Under a quantitative nondegeneracy condition, motivated by calibrated material regimes, the coupled inverse problem reduces to a single inverse source problem. Applying the microlocal and Carleman framework of Stefanov and Uhlmann to this reduced problem, we establish global uniqueness and Lipschitz stability under geometric conditions involving strictly convex foliations and geodesic visibility. We also obtain partial-data results, proving local uniqueness and Lipschitz stability in compact visible regions where the constitutive nondegeneracy condition holds.

This paper concerns neutral inclusions for imperfect bonding of high-conductivity type. An inclusion, which is a bounded domain, is said to be of imperfect bonding of high-conductivity type if the flux is discontinuous along its boundary while the potential is continuous. The inclusion is neutral to a uniform field if the presence of the inclusion does not perturb the field outside the inclusion. It is known that ellipses and ellipsoids can be neutral to all uniform fields by introducing a proper imperfect bonding coefficient on boundaries. The purpose of this paper is to prove the converse. We prove that if an inclusion of imperfect bonding of high-conductivity type is neutral to all uniform fields, then it is an ellipse or an ellipsoid. The neutrality condition is given by existence of the solution to a certain differential equation on the boundary surface and the main result is proved by converting the neutrality condition into an algebraic boundary identity characterizing ellipses and ellipsoids.

We establish a quantitative adiabatic estimate for a class of finite-dimensional non-Hermitian Schrödinger dynamics. The original non-Hermitian Hamiltonian is assumed to be diagonalizable with real spectrum and non-crossing eigenvalues. We then construct a dynamically compatible time-dependent metric operator, and its positive square root defines a Dyson map. The associated Dyson-transformed Hamiltonian is Hermitian. When this Dyson-transformed Hamiltonian satisfies the Hermitian adiabatic assumption, the standard resolvent-projection method gives an explicit adiabatic estimate in the Hermitian representation. Pulling this estimate back through the Dyson map gives an approximation in the original non-Hermitian representation. The resulting Dyson-pulled-back projections are then compared with the spectral projections of the original non-Hermitian Hamiltonian by using a contour-resolvent estimate. The final bound contains two contributions: the pulled-back Hermitian adiabatic error and the projection-comparison error. A two-level non-Hermitian model is presented to illustrate the hypotheses of the theorem and the two contributions appearing in the final estimate.

We study the linearized motion of a spherical rigid body immersed in an incompressible viscous fluid occupying the whole space, under Navier slip boundary conditions with friction. While semigroup regularity and decay estimates are well understood for the no-slip case, such results have been limited to bounded domains in the slip setting. By exploiting the geometric regularity of the sphere, we establish strong $L^q$-regularity, bounded analyticity, and sharp $ L^q-L^r$ estimates for the corresponding fluid-structure operator. These results would provide a foundation for constructing strong $W^{2,1}_q$ solutions and Kato-type solutions to the corresponding nonlinear system.

This paper aims to establish the global regularity and large time behavior of solutions to the three-dimensional (3D) compressible magnetohydrodynamics (MHD) equations with vanishing vertical magnetic resistivity in the upper half-space with no-slip boundary condition on velocity and perfectly conducting boundary condition on magnetic field. By exploiting anisotropic Sobolev inequalities and elaborated estimates, we are able to achieve global-in-time uniform regularity estimates of solutions, which are independent of small vertical resistivity coefficient $\varepsilon$. These uniform global regularity estimates allow us to pass to the limit as \(\varepsilon \to 0\) and obtain the convergence to the corresponding MHD system without vertical magnetic resistivity globally in time. Moreover, the $H^{1}\left(\mathbb{R}_{+}^{3}\right)$ decay rates of solutions to the original system are also derived based on the detailed analysis on semigroup of the related linearized operator in half space and the uniform energy estimates achieved. In contrast to the decay estimates for the limit system obtained by Gao and Xie \cite{JDE}, the present decay estimate exhibits a slower rate, attributable to the absence of higher-order normal derivative estimates, which results from the occurrence of boundary layers. Finally, we combine both sets of regularity and decay results to rigorously prove an explicit time-uniform $L^2$ convergence rate of order $\varepsilon^{\frac{1}{4}}$ for the vanishing vertical magnetic resistivity limit process.

Let $0<a<1$ and let $u_0$ be a $C^1$, divergence-free, $(-a)$-homogeneous vector field on $\mathbb{R}^2\setminus\{0\}$. We construct a forward-in-time self-similar solution of the two-dimensional incompressible Euler equations, \[ u(t,x)=t^{-\frac{a}{1+a}} U\left(\frac{x}{t^{{\frac{1}{1+a}}}}\right), \] with initial datum $u_0$. No smallness or sign assumption is imposed on the initial datum. The construction is a vanishing-dissipation limit of hypodissipative self-similar profiles built directly in the critical vorticity space. The main estimate is a uniform critical Lorentz bound $\|\operatorname{curl} U\|_{L^{\frac{2}{1+a},\infty}(\mathbb{R}^2)}$. The resulting Euler solution $u$ belongs to $C([0,\infty);L^2_{\mathrm{loc}}(\mathbb{R}^2))$ and has vorticity uniformly bounded in the critical space $L^{\frac{2}{1+a},\infty}(\mathbb{R}^2)$. Its velocity converges strongly to $u_0$ in $L^2_{\mathrm{loc}}$, while its vorticity converges weak-star to $\omega_0=\operatorname{curl} u_0$ in $L^{\frac{2}{1+a},\infty}$ as $t\downarrow0$.

We introduce an infinite family of mean-value formulas (exact and asymptotic) given in terms of linear combinations of iterated means. We prove that the mean-value formulas in this family characterize real-valued polyharmonic functions of finite order, and that a simple algebraic condition partitions the family into equivalence classes according to the order of polyharmonicity. Our key results include strong converses to the mean-value properties – locally integrable functions satisfying a mean-value property in the family are polyharmonic – and a regularity result – locally integrable functions satisfying a mean-value property in the family, whether exact or asymptotic, are smooth.

We study bounded entire solutions $u:\mathbb{R}^2\to \mathbb{R}^2$ that minimize the Allen-Cahn functional \begin{equation*} J(u,\Omega)=\int_\Omega \left(\frac12 |\nabla u|^2+W(u)\right)\,d\mathbf{x}, \end{equation*} with the $D_3$-invariant triple-well potential \begin{equation*} W(u_1,u_2)=|u|^4+2u_1u_2^2-\frac23 u_1^3-|u|^2+\frac23. \end{equation*} We obtain a complete classification of entire minimizing solutions. In particular, when $u$ has a triple-junction structure at infinity, up to translation and orthogonal change of coordinates, $u$ has the explicit profile \begin{equation*} u_*(\mathbf{x})=\sum_{i=1}^3 \frac{e^{\sqrt2 a_i\cdot \mathbf{x}}}{\sum_{j=1}^3 e^{\sqrt2 a_j\cdot \mathbf{x}}}a_i. \end{equation*} We also demonstrate that the solutions obtained by minimizing within the $D_3$-equivariant class coincide with $u_*$. The key ingredient is a calibration identity arising from the special algebraic structure of $W$.

We establish the global well-posedness of the Cauchy problem for the reverse space-time nonlocal Fokas-Lenells equation with the weighted Sobolev initial data $q_0(x)\in H^{3}(\mathbb{R}) \cap H^{2,1}(\mathbb{R})$ on the line. We develop the inverse scattering transform formulated via the associated Riemann-Hilbert problems to study this issue. A spectral uniformization transform is introduced to resolve the singular behavior inherent in the KN-type negative flow spectral problem. Owing to the reverse space-time reduction, reflection coefficients no longer satisfy the usual Hermitian conjugation symmetry, and the coercivity of the jump matrix is therefore not available a priori. The quantitative smallness condition on the initial data yields uniform bounds on the reflection coefficients and ensures the uniform positive definiteness of the Hermitian part of the associated jump matrix. The resulting coercivity allows us to establish the bounded invertibility of the associated singular integral operator through a Fredholm and vanishing-lemma argument. Under this condition, we prove an $L^{2}$-Sobolev bijective correspondence between the potential and scattering data, exclude spectral singularities on continuous spectra, and obtain the global existence and uniqueness of solutions. Moreover, the associated solution map is Lipschitz continuous on the admissible initial-data class.

We study the one-dimensional cubic nonlinear Schrödinger equation with a repulsive delta potential and a localized inhomogeneous coefficient. We prove small-data modified scattering and construct the associated vector-valued modified scattering map, whose two components encode the coupling of the frequencies $\xi$ and $-\xi$ induced by the point interaction. We show that this map determines both the strength of the delta potential and the inhomogeneous coefficient. Quantitatively, the delta strength is recovered with Lipschitz stability, while the inhomogeneous coefficients satisfy a Hölder stability estimate. To our knowledge, these are the first recovery and stability results for a modified scattering map in the presence of an external potential.

We study the uniform $L^\infty(\Omega)$ a priori boundedness of positive weak solutions to the fractional semilinear Dirichlet problem $(-\Delta)^s u = f(u)$ in a bounded, convex, $C^{1,1}$ domain $\Omega \subset \mathbb{R}^N$ with homogeneous exterior condition $u\equiv 0$ in $\mathbb{R}^N\setminus\Omega$. We consider slightly superlinear nonlinearities of the form $f(t) = t^q L(t)$, where $1 \le q \le \frac{N+2s}{N-2s}$ and $L$ is a slowly varying function. Although uniform estimates are well-established in the strictly subcritical regime $q < \frac{N+2s}{N-2s}$, the slightly subcritical case, $q = \frac{N+2s}{N-2s}$, is highly challenging due to the potential formation of bubbling profiles. In this work, we isolate a structural condition on the slowly varying perturbation, namely $$ \lim_{t \to \infty} \frac{t \, |L'(t)|}{L^{\frac{N}{2s}}(t)} = \infty, $$ which acts as an asymptotic barrier that prevents mass concentration. Under this assumption, we establish global uniform $L^\infty(\Omega)$ bounds for positive solutions, significantly expanding the class of known nonlinearities for which such estimates hold.

Essentially non-oscillatory (ENO) reconstruction provides a key mechanism for designing high-order entropy-stable schemes for hyperbolic conservation laws, with its sign property ensuring nonnegative local dissipation for a prescribed entropy. However, two fundamental questions concerning convergence remain open: whether this dissipation provides the coercivity required for weak-BV compactness, as posited by the ENO–TV conjecture, and whether entropy stability transfers from the prescribed entropy pair to additional pairs. This paper resolves the ENO–TV conjecture by establishing a sharp parity dichotomy: it holds if and only if the reconstruction order $k=2$ or $k$ is odd, and fails for all even orders $k\ge 4$.The key to our proof is a localization principle that eliminates dependence on nonlinear adaptive stencil selection, establishing a two-sided equivalence between ENO dissipation and a canonical finite-difference functional. For odd orders, the conjecture is proved via a hidden quadratic energy and novel discrete Gagliardo–Nirenberg inequalities. For even orders $k\ge 4$, ENO null modes, on which ENO dissipation vanishes, yield counterexamples that disprove the conjecture. This dichotomy extends to quasi-uniform meshes, but for every $k\ge 2$, the conjecture can fail on non-quasi-uniform meshes. Addressing the above second open question, we discover on ENO null modes that local entropy transfer is governed by the first cohomology of a unipotent shift. Using apolar duality and binary covariants, we compute the dimensions of the associated cohomology subspaces and prove that smooth local entropy transfer encounters generic obstructions for every $k\ge 4$. By revealing how ENO null modes link global coercivity and local entropy compatibility, this work provides a structural foundation for the compactness and convergence analysis of high-order entropy-stable discretizations.

The existence of non-trivial solutions of homogeneous partial differential equations with prescribed support properties plays a fundamental role in the theory of linear partial differential operators. In this article, we establish the existence of smooth zero solutions with partially bounded supports for a broad class of constant coefficient partial differential operators. As applications, we obtain new geometric results concerning solvability and approximation for partial differential equations. In particular, we derive geometric characterizations of $P$-convexity for supports for a large class of non-elliptic operators, thereby extending classical geometric ideas of Hörmander. By a theorem of Malgrange, $P$-convexity for supports is equivalent to the surjectivity of the differential operator on spaces of smooth functions and, more generally, on local subspaces of distributions of finite order. As a further consequence, we prove that the kernel of every surjective semi-elliptic operator satisfies condition ($\Omega$) implying parameter dependence results. We also investigate Runge-type approximation phenomena. We obtain geometric characterizations of Runge pairs for smooth functions, distributions, and spaces of smooth Whitney jets. Finally, we develop a general framework for Runge approximation for square systems of constant coefficient partial differential equations. As applications, we recover by alternative methods known Runge approximation results for Beltrami fields and for the three-dimensional unsteady Stokes system, and we complement them by corresponding approximation theorems in spaces of smooth Whitney jets. In particular, this yields approximation up to the boundary for solutions on suitable domains with Hölder continuous boundary.

In this paper, we study the Hardy-Hénon parabolic equation \begin{equation*} \partial_t u=\Delta u+a|x|^{-\gamma}|u|^{\beta}u, \qquad t>0,\; x\in\mathbb{R}^{n}\setminus\{0\}, \quad \beta>0,\; a\in\mathbb{R}, \end{equation*} under suitable assumptions on the parameter $\gamma$. We establish the local and global well-posedness of mild solutions in homogeneous Herz spaces.

We prove a desingularization theorem for analytic vortex sheets of the 3D incompressible Euler equations. Starting from an analytic solution of the corresponding Birkhoff-Rott system, we construct, for every sufficiently small thickness parameter $\varepsilon>0 $, an exact Euler vorticity supported on a tubular neighborhood of width $ O(\varepsilon) $ around the sheet, and defined on a time interval that does not shrink to 0 as $\varepsilon \to 0$. We show that, as $\varepsilon \to 0$, these vorticities converge, in the sense of distributions, to the prescribed vortex sheet. In particular, we conclude that analytic 3D vortex sheet motions arise as limits of exact Euler flows with lifespan bounded from below independently of $ \varepsilon $. The proof hinges on the study of vorticities defined in terms of a time-dependent foliation by almost parallel surfaces and of divergence-free vector fields tangent to these surfaces.

We show that, for almost-minimizers of the Griffith energy in the plane, the complement of the crack is locally covered by a finite number of John domains where the displacement satisfies Hölder-type estimates. As a consequence, we derive the integrability of the full gradient and the finiteness of the traces along the crack. In particular, we show that any Griffith almost-minimizer in dimension two locally belongs to the space SBV^2.

We consider stabilisation of abstract boundary control systems and controlled partial differential equations with general unknown input disturbances and unmodeled nonlinearities at the input. We utilise the active disturbance rejection control approach to design a controller which rejects the input disturbance and achieves stability and external well-posedness of the closed-loop system for a class of boundary control systems with collocated inputs and outputs. We apply our main results to design controllers for one-dimensional wave and heat equations.

In this paper, we investigate the long-time asymptotics for the solution of the Cauchy problem of the defocusing Hirota equation on a finite-genus algebro-geometric background in the whole $(x,t)$-half-plane, whose method is mainly based on a Riemann-Hilbert (RH) formulation and Deift-Zhou nonlinear steepest descent method. The critical values of the phase function in the associated RH problem divide the space-time plane into four regions, in which the leading-order term is given by a phase-shifted finite-genus algebro-geometric solution. The subleading behavior depends on the region: the correction is of order $t^{-1/3}$ and is governed by a Painlevé-XXXIV model RH problem in the transition regions; the leading radiation is of order $t^{-1/2}$ in the Zakharov–Manakov region; and the error is $O(t^{-1})$ in the fast-decay region. These results can also be extended to other higher-order members of the AKNS hierarchy.

The paper introduces a stochastic model for the spontaneous formation of traffic waves on a highway. This is formulated in terms of a conservation law with discontinuous, gradient-dependent flux. In an unstable regime, the non-uniqueness of solutions allows for the emergence of $N$-shaped spikes in the traffic density, at random points and times. Bounds are proved on the expected value of the total variation of the random solution and on the expected number of shocks. Further bounds are obtained on the average velocity and on the expected average acceleration of cars, along a given stretch of highway. Finally, it is proved that the Markov process, whose paths are random solutions to the conservation law, admits a unique stationary probability distribution and is ergodic.

In this paper, a multivariate Mittag–Leffler-type function arising in the theory of fractional differential equations with several fractional parameters is investigated. New Hankel contour integral representations are derived for the three-variable Mittag–Leffler function, and complete asymptotic expansions are established in different sectors of the complex plane. The proposed approach is based on the classical Hankel integral representation of the reciprocal Gamma function together with suitable contour transformations. Furthermore, the corresponding integral representations and asymptotic expansions are established for the multivariate Mittag-Leffler function with an arbitrary number of variables. The obtained formulas extend several known results for one- and two-variable Mittag–Leffler functions and provide useful analytical tools for the qualitative analysis of fractional differential equations involving multiple fractional derivatives.

Let $n\equiv 3\ (\mathrm{mod}\ 4)$ and set $p=\frac{n-1}{2}$. On an oriented Riemannian $n$-manifold we consider the (middle-degree) curl operator, $\mathrm{curl}:*\mathrm{d}:\Omega^{p}\rightarrow\Omega^{p}$, and the associated conformally invariant Sobolev quotients on $(\mathbb{S}^n,g_{\mathrm{st}})$, \[ J_1(\alpha)=\frac{\big(\int|\mathrm{curl}\alpha|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\int\langle\mathrm{curl}\alpha,\alpha\rangle\,\mathrm{dV}}, \qquad J_2(\alpha)=\frac{\big(\int|\mathrm{curl}\alpha|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\inf_{\phi}\big(\int|\alpha-\mathrm{d}\phi|^{\frac{2n}{n-1}}\,\mathrm{dV}\big)^{\frac{n-1}{n}}}. \] Killing $p$-forms and their conformal images form a natural family of critical points for both functionals, analogous to the Aubin-Talenti family in the classical Sobolev inequality. We prove a quantitative local stability estimate for $J_1$ around this family, which in particular implies that every such form is a strict local minimizer in the conformally invariant space $W^{1,\frac{2n}{n+1}}$. In contrast, we show that these critical points are unstable for $J_2$ (and for related conformally invariant quotients), yielding a strict upper bound for the sharp constant of the $J_2$ inequality. By conformal invariance, the results on $\mathbb{S}^n$ transfer naturally to $\mathbb{R}^n$.

We investigate time-periodic dynamics in a forager-exploiter system with a taxis cascade under both homogeneous and heterogeneous resource environments. The model describes the interactions among foragers, exploiters, and environmental resources, where foragers move toward higher resource densities while exploiters aggregate toward regions with higher forager densities. Our results show that different resource renewal mechanisms shape periodic dynamics in fundamentally different ways. Precisely, for time-periodic resource renewal rates, we establish the existence of positive time-periodic solutions for any positive renewal rate and further prove their global stability under suitable conditions on the parameters. In contrast, for homogeneous environments, only large resource renewal rates can destabilize the constant steady state through the Hopf bifurcation, thereby generating non-constant time-periodic solutions. Interestingly, for spatially heterogeneous and temporally homogeneous environments, our numerical simulations indicate that spatial heterogeneity exhibits opposing effects on periodic dynamics depending on total resource availability. When resources are sufficiently abundant, spatial heterogeneity tends to suppress the emergence of temporal oscillations, whereas when resources are relatively scarce, it may instead promote oscillatory behaviors, and sufficiently concentrated local resource supplies can trigger local or even global temporal oscillations. These findings reveal a delicate interplay among resource renewal mechanisms, resource availability, and spatial heterogeneity in shaping dynamics behavior.

We investigate Hartree-type equations driven by a nonlocal operator $\mathcal{L}_\mu$, defined as a superposition of fractional Laplacians through a signed Borel measure $\mu$. Under Berestycki-Lions type assumptions, we prove the existence of a Mountain Pass solution and show that its energy level coincides with the minimum on the Pohozaev manifold. We also establish the boundedness of non-negative solutions. The proof of this fact requires a careful use of the Sobolev embedding in the iterative argument and a delicate treatment of the integrals involved in the estimates, as well as a Kato-type inequality in our general setting. Finally, we establish a general Pohozaev identity for solutions under a suitable summability assumption.

Classifying the spherical metrics on a torus $E_\tau$ with $4\pi$ conic angle at each half period point\, ${\omega_k}/{2}, k=0,1,2,3$\, is equivalent to classify solutions of the following curvature equation \begin{align}\label{eq0731093154} \Delta u+e^u=4\pi\sum_{k=0}^3\delta_{\frac{\omega_k}{2}}\text{\ on\ }E_\tau \end{align} where $\tau\in \mathbb{H}:=\{z\in \mathbb{C}\mid \mathrm{Im} \, z>0\}$ and $\delta_p$ is the Dirac measure at $p\in E_\tau$. By constructing a multiple Green function $$G_2(z_1, z_2;\tau):=G(z_1-z_2;\tau)-\frac{1}{2}\sum_{j=0}^3\left(G(z_1-\frac{\omega_j}{2};\tau)+G(z_2-\frac{\omega_j}{2};\tau)\right), $$ in terms of the Green function $G(z;\tau)$ on $E_\tau$, we classify the solutions of (\ref{eq0731093154}) into two types: special and non-special. Furthermore, we obtain the following conclusion about the solutions of (\ref{eq0731093154}): \begin{enumerate} \item any special solution is an even function and the set of special solutions is isomorphic to $SL(2,\mathbb{C})/SU(2)$ for all $\tau\in \mathbb{H}$. \item a non-special solution exists if and only if $\tau\in \mathcal{E}$. Moreover, if $\tau\in \mathcal{E}$, then there are six one-parameter families of nonspecial solutions. \end{enumerate} where $$\mathcal{E}:=\left\{\tau\in \mathbb{H}\mid G(z;\tau) \,\,\text{has exactly 5 critical points.}\right\}.$$ The set $\mathcal{E}$ is completely determined in \cite{CLW2018, Lin}, which is a union of countable many open triangular domains. As a byproduct, we completely determine and classify the critical points of $G_2$ and then obtain the degeneracy criterion of critical points for $G_2$, which may be of independent interest.

The hydrostatic balance is a fundamental equilibrium state in stratified fluids and plays a central role in geophysical fluid dynamics. Understanding its stability under incomplete dissipation is a longstanding challenge, since anisotropic diffusion alone is generally insufficient to control the nonlinear evolution and no robust stabilizing mechanism is known for the corresponding anisotropically dissipative Navier–Stokes equations. In this paper, we investigate the two-dimensional Boussinesq equations on $\mathbb{R}^2$ with only horizontal dissipation near the hydrostatic equilibrium $(U,\Theta)=(0,x_2)$. We show that the velocity–temperature coupling generates internal gravity waves whose dispersive decay, together with the horizontal dissipation, provides an effective stabilizing mechanism that compensates for the complete absence of vertical dissipation. This identifies a mechanism by which dispersive wave propagation restores stability in an incompletely dissipative fluid system. For sufficiently small initial perturbations in $H^k(\mathbb{R}^2)\cap W^{3,1}(\mathbb{R}^2)$ with $k\ge14$, we establish the global existence and uniqueness of classical solutions together with explicit anisotropic, componentwise large-time decay rates for the velocity and temperature, including faster decay of the vertical velocity.

This work is devoted to the study of traveling wave solutions of reaction-diffusion systems, where the diffusion rate of~$v$, the second component, depends on~$u$, the first component. Such systems arise in prey-predator models, where the predator~$v$ is actively hunting its prey~$u$, or in epidemiological models where the disease induces erratic behavior, for example rabies. This results in a quasilinear coupling in the highest-order term of the equation for~$v$. From the theoretical point of view, we fully classify traveling wave solutions when the first component does not diffuse, in which case the problem can be reduced to a nonlinear scalar equation. The case where both components diffuse is investigated numerically.

We study the spreading dynamics for two-species diffusive competition systems in shifting environments caused by climate changes. Our main goal is to derive conditions for extinction and persistence of each species in the case of a strong-weak competition. Depending on the pace of the climate change, but also on whether the strong competitor is faster or slower, we will uncover dramatically different outcomes in the asymptotic behavior of solutions. For instance, it may be that a weak and fast competitor survives while a strong and slow one does not. Furthermore, we find some parameter regime where the outcome depends not only on the climate change speed, but also on both species' resilience to it, and even on the initial populations' distributions. Our results show how strikingly complex the effect of climate change may be on population dynamics as soon as several species are considered.