Papers

We construct an explicit strict Lyapunov function for the one-dimensional linear wave equation with a locally distributed damping and Dirichlet boundary conditions.

So far the global well-posedness of strong solutions for the 3D non-diffusive Boussinesq system with large initial data remains a remarkable open problem. In this paper, we solve this problem in the regime of large Prandtl number. More precisely, we prove the global existence and uniqueness of strong solution for this 3D Boussinesq system associated with initial data $(u_0,\theta_0)\in H^{\frac{1}{2}}(\mathbb{R}^3) \times (L^1\cap L^s(\mathbb{R}^3))$ with $s>3$, provided that the Prandtl number is sufficiently large (the threshold depends only on a scale-invariant norm of $(u_0,\theta_0)$); moreover, for the non-constant temperature patch initial data, we establish the global persistence of $C^{1,\gamma}$, $W^{2,\infty}$, and $C^{2,\gamma}$ ($0<\gamma<1$) boundary regularity of the evolved temperature patch, with corresponding estimates uniform in the large Prandtl number regime. Furthermore, we rigorously justify the limit as the Prandtl number tends to infinity and show that the patch solution of the 3D Boussinesq system converges to the unique patch solution of the 3D Stokes-transport system, and that the patch boundary regularity in $C^{1,\gamma}$, $W^{2,\infty}$, and $C^{2,\gamma}$ is preserved globally in time. In particular, our result for the 3D Stokes-transport system can be viewed as the 3D analogue of the main result in Grayer II [ARMA 2023] concerning 2D Stokes-transport system.

This article is devoted to the inverse problem of determining the zeroth- and first-order coefficients, depending on both the time and space variables, in a parabolic equation from partial boundary measurements of the flux generated by Dirichlet excitations. More precisely, we establish the unique determination of a time-dependent convection term and potential from the partial Dirichlet-to-Neumann map associated with the corresponding parabolic equation, where the Neumann measurements are restricted to the portion of the boundary illuminated from a point located outside the closure of the domain. Our main objective is to extend to parabolic equations several observations that have thus far been confined to the elliptic setting and to exploit these properties to recover a general class of coefficients depending on both time and space variables. To achieve this, we develop a suitable class of special solutions to the parabolic equation, introduce a new family of Carleman estimates with nonlinear weights, and combine these tools with partial data results for the X-ray transform. The principal difficulty of the present work stems from the fact that, unlike the existing literature on partial data inverse problems for parabolic equations, both the phase of the special solutions and the weight of the corresponding Carleman estimates are nonlinear. As a byproduct of our analysis, we establish a new class of Carleman estimates for parabolic equations with nonlinear weights, which may be viewed as the parabolic analogue of the notion of limiting Carleman weights in the elliptic setting.

We study a nonlinear tensorial wave-Dirac system on $(1+3)$-dimensional asymptotically flat spacetimes as a semilinear model motivated by the Maxwell-Dirac and Einstein-Dirac systems. The purpose of this model is to isolate the interaction between the null geometry of antisymmetric tensor fields and the intrinsic first-order geometry of the Dirac equation while avoiding the derivative loss mechanism of the Einstein equations and the gauge structure of the Maxwell equations. Our analysis preserves the first-order nature of the Dirac equation throughout the nonlinear argument. The Dirac current provides the fundamental energy identity, while quantitative spacetime estimates are obtained from the wave equation arising from the squared Dirac operator. Combining these ingredients with integrated local energy decay estimates and $r^p$-weighted energy hierarchies, we establish a coupled energy method for the tensorial and spinorial components. A key observation is that the Clifford algebra is compatible with the null decomposition of antisymmetric tensor fields and excludes the most singular nonlinear interactions. As a consequence, we establish the global existence of small-data solutions together with quantitative weighted energy and decay estimates. Remarkably, combining the null structure with dyadic argument, weak decay such as $t^{-\frac12-\delta}$ is sufficient to obtain nonlinear stability of the system, in the spirit of \cite{DHRT}. We expect that the geometric ideas developed in this paper provide a useful starting point for the study of more general nonlinear Dirac systems, including the Maxwell-Dirac and Einstein-Dirac equations.

The purpose of this note is twofold. First, we prove quantitative estimates for the Ambrosio-Trevisan commutator which implies propagation of logarithmic Sobolev regularity. Second, we prove a similar estimate for the DiPerna-Lions commutator by relying on quantitative differentiation.

Let $(X^{n+1},g_+)$ be a Poincaré–Einstein manifold with conformal infinity $(M^n,[h])$ of positive Yamabe constant $Y(M,[h])$. Then a relative comparison inequality \[ \frac{Y_1(X,M,[\bar g])}{Y_1(\mathbb{S}^{n+1}_+,\mathbb{S}^n,[g_{\mathbb{S}_+^{n+1}}])} \geq \left(\frac{Y(M,[h])}{Y(\mathbb{S}^n,[g_{\mathbb{S}^n}])}\right)^{\frac{n}{n+1}} \] holds for the type-I Escobar–Yamabe compactification $Y_1(X,M,[\bar g]$, with equality if and only if $\left(X, g_{+}\right)$ is isometric to the hyperbolic space $\left(\mathbb{H}^{n+1}, g_{\mathbb{H}^{n+1}}\right)$. This confirms a conjecture raised by Sun-Yung A. Chang.

In this paper, we investigate the global well-posedness of strong solution to a model recently derived by Constantin-Johnson \cite{CaJo} which describes the nonlinear wave propagation in the troposphere, especially for the morning glory cloud. Assuming that the initial velocity $v_0\in H^1$ and the thermodynamic forcing term $K\in L^2(0,T;L^2)$, we show that there exists a unique global strong solution to the initial boundary value problem of this system. Similar result was only known before for sufficiently small initial data in the existing literature.

We prove norm inflation, in the sense of strong ill-posedness, for the two-dimensional Boussinesq system in supercritical Besov spaces. For the inviscid system, norm inflation holds in \(\dot B^\beta_{p,q}(\mathbb R^2)\times \dot B^\beta_{p,r}(\mathbb R^2)\) for \(\beta\neq0\), \(1<p\leq\infty\), \(1\leq q,r\leq\infty\), and \(-2<\beta-\frac{2}{p}<1\). For the fully dissipative system, the same conclusion holds in the range \(-2<\beta-\frac{2}{p}<-1\). In both cases, the results cover almost all supercritical Besov spaces satisfying the local integrability condition. Norm inflation occurs in the density component \(\rho\), while the velocity component \(u\) remains bounded. In the fully dissipative case, the inflation space is supercritical for \(u\), but subcritical for \(\rho\) with respect to its own scaling. This is not a contradiction: the density is transported by a velocity field in a supercritical regime, and this transport mechanism is precisely what produces norm inflation in \(\rho\).

We study the existence and asymptotic behaviour of normalized solutions to the following pseudo-relativistic Schrödinger equation with logarithmic nonlinearity \[ (\sqrt{-\Delta+m^2 }-m )u+i(v\cdot \nabla )u+\lambda u = u\log|u|^2+|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N, \] under the mass constraint \[ \|u\|_2^2=a, \] where $m,a>0$, $2<p\le\frac{2N}{N-1}$ with $N\ge 2$, $v\in \mathbb{R}^N$ is the travelling velocity with $|v|<1$, and $\lambda\in\mathbb{R}$ appears as Lagrange multiplier, as minima of the corresponding energy on the constraint. By applying variational method, we first provide a complete classification of the existence and nonexistence of such minima. In particular, for the mass-critical case $p=2+\frac{2}{N}$, we show that there exists a constant $a^\ast_v$ which is a threshold for the existence. Based on this, we analyse the blow-up behaviour of such minimizers as $a$ approaches $a^\ast_v$ from below. Finally, we investigate the limiting profiles of minimizers to problem when $\lim\limits_{n\to\infty}a_n=a_0\in(0,+\infty)$ with $\{a_n\}\subset(0,+\infty)$ in the mass-subcritical case $2<p<2+\frac{2}{N}$ and $\lim\limits_{n\to\infty}a_n= a_0\in(0,a^\ast_v)$ with $\{a_n\}\subset(0,a^\ast_v)$ in the mass-critical case $p=2+\frac{2}{N}$, respectively.

We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.

We provide a constructive existence proof for overhanging periodic gravity water waves with constant vorticity. By introducing a conformal mapping formulation, we parameterize the fluid domain to accommodate multi-valued surface heights, bypassing coordinate singularities that arise in traditional domain-flattening frameworks. We operate at fixed, macroscopic physical parameters and employ a global bifurcation framework, supplemented by a computer-assisted proof based on the Newton-Kantorovich theorem, to constructively prove the existence of the branch of exact solutions. Crucially, this allows us to provide the first rigorous existence proof of overhanging periodic gravity water waves obtained along a finite-depth global bifurcation branch from the flat state at fixed $O(1)$ physical parameters. Moreover, our analysis confirms the existence of a global solution branch that transitions to overhanging profiles and resolves the conjecture of Constantin, Strauss, and Varvaruca regarding the topological termination of this branch via physical self-intersection at the trough line.

Let $G(z)$ be the Green function on the flat torus $E_{\tau}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ with the singularity at $0$. Lin and Wang (Ann. Math. 2010) proved that $G(z)$ has at most one pair of nontrivial critical points. This is the third of a series of papers to study the sum of two Green functions which can be reduced to $G_p(z):=\frac12(G(z+p)+G(z-p))$. We study how the geometry of the torus and the location of singularities $\pm p$ affect the structure of critical points of $G_p(z)$. In Part I \cite{CFL}, we proved that $G_p(z)$ has at most three pairs of nontrivial critical points for all tori. In Part II \cite{CFL-II} (Proc. Lond. Math. Soc. 2026), we studied the important case that $E_{\tau}$ is a rectangular torus. In this paper, first we prove that if $G_p(z)$ has three pairs of nontrivial critical points, then critical points are all non-degenerate. Secondly, we study the other important but more challenging case that $E_{\tau}$ is a rhombus torus, by developing different approaches from \cite{CFL, CFL-II}. As applications, we show that the curvature equation $\Delta u+e^{u}=4\pi(\delta_p+\delta_{-p})$ on $E_{\tau}$ has exactly either $0$, $1$ or $2$ even axisymmetric solutions and each number really occurs.

We prove a quantitative propagation of chaos estimate and a central limit theorem for the particle system associated with a class of degenerate kinetic McKean–Vlasov SDEs with external drifts and singular interaction kernels in Kato's class. In particular, the interaction kernel can be in the mixed $L^q_tL^{p_v}_vL^{p_x}_x$-space, where $\frac2q+\frac{3d}{p_x}+\frac d{p_v}<1$. For the associated $N$-particle system, we obtain a path-space relative entropy bound of order $k/N$ for the first $k$ particles, assuming only entropic chaoticity of the initial data. The key ingredients are kinetic Krylov–Khasminskii estimates and a conditional Hilbert-space subgaussian estimate for empirical interaction fields. For the CLT, we also prove a Berry–Esseen-type bound for finite-dimensional projections.

This is the third part in a series of papers developing a new framework to study the local structure of $n$-dimensional area-minimizing rectifiable currents $T$ of codimension $\geq 2$. Parts I and II introduced an intrinsic frequency function for $T$ – planar frequency – and used its monotonicity properties, among other things, to establish that ${\mathcal H}^{n-2}$-a.e. branch point is a rapid-decay branch point where the planar frequency is at least $1 + \alpha$. This paper analyses branch points of planar frequency $\neq 2$. It establishes: (1) higher order asymptotics: at ${\mathcal H}^{n-2}$-a.e. such point, the current admits an expansion of finite order $>1$, with precise decay estimates for the remainder term; (2) branch set decomposition: the set of such branch points locally decomposes into finitely many pairwise disjoint, locally $n-2$ rectifiable sets (of locally finite measure); (3) topological control: near any branch point satisfying a specific planar-frequency criterion, the support of $T$ is homeomorphic to an $n$-dimensional disk and admits a $C^{1, \mu}$ parametrization. (Classical complex algebraic examples show that when this frequency criterion fails, the current need not be locally homeomorphic to an $n$-disk). The work here (as well as in parts I & II) avoids the use of center manifolds – a technically demanding foundational component of the classical Almgren framework – and uses instead intrinsic geometric arguments based on the monotonicity formula for planar frequency. In part IV, a center manifold is utilised to analyse planar frequency 2 points, where the center manifold becomes necessary and geometrically canonical, satisfying additional simplifying properties. Reduced reliance on center manifolds in our framekwork is necessitated by the structural results it establishes for $T$.

This article investigates positive supersolutions of the fully nonlinear elliptic system \[ \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)+|\nabla u|^{q} \geq\lambda_{1}f_{1}(v)~~\text{in}~~\mathbb{R}^{n}\setminus B_{R_0},\\ -\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}v)+|\nabla v|^{q} \geq\lambda_{2}f_{2}(u)~~\text{in}~~\mathbb{R}^{n}\setminus B_{R_0}, \end{cases} \] where $q>1,\lambda_{1},\lambda_{2}>0,$ and the nonlinearities exhibit power-type behaviour either near the origin or at infinity. Introducing the effective dimension $\widetilde n_{+}=\frac{\lambda}{\Lambda}(n-1)+1$ associated to extremal Pucci operator, we identify the critical exponent $q_{c}=\frac{\widetilde n_{+}}{\widetilde n_{+}-1},$ which governs the qualitative behaviour of positive supersolutions. Using this framework, we establish sharp Liouville-type nonexistence theorems in exterior domains and determine optimal nonexistence regions through the interaction between the gradient exponent $q,$ the effective dimension $\widetilde n_{+}$ and the nonlinear couplings. In the prototype case $f_{1}(t)=t^{p_{1}},$ $f_{2}(t)=t^{p_{2}}$ the obtained conditions are shown to be optimal. The analysis is carried out under the natural regularity assumption $u,v\in W^{2,p}_{\mathrm{loc}}(\mathbb{R}^{n}\setminus B_{R_0}),$ for $p>n,$ which is the regularity available for fully nonlinear uniformly elliptic equations, rather than within a classical $C^{2}$ framework. Our results provide the fully nonlinear Pucci analogue of the Liouville theory for semilinear elliptic systems involving nonlinear gradient terms and reveal the fundamental role of the effective dimension in determining the critical nonexistence thresholds.

We investigate a radially symmetric initial-boundary value problem whose separation of variables leads to a regular self-adjoint Sturm-Liouville problem with explicitly determined coefficients and a positive weight function. The resulting Sturm-Liouville operator is defined by a special geometry-induced coefficient pair that gives rise to a nonstandard weighted spectral structure. The associated radial operator possesses a discrete real spectrum and a complete orthonormal system of eigenfunctions in the corresponding weighted Hilbert space, yielding an exact spectral representation of the evolution problem. An exact transcendental spectral equation governing the eigenvalues is derived. To analyze the high-frequency regime, the Liouville-Green transformation is applied directly to the radial Sturm-Liouville equation. This yields explicit asymptotic formulas for the eigenvalues and eigenfunctions together with corresponding Liouville-Green quasimodes. Their weighted orthogonality properties and asymptotic completeness are established within the spectral framework of the exact operator. The resulting asymptotic spectral data are used to construct approximate solutions of the original boundary-value problem and to obtain error estimates for the spectral reconstruction. Numerical computations of the exact Sturm-Liouville spectrum show excellent agreement with the Liouville-Green predictions, thereby validating the asymptotic quantization formula and confirming the accuracy of the proposed spectral approximation.

We study multidimensional semilinear stochastic evolution equations driven by multiplicative noise and subject to rough initial data. The drift and diffusion coefficients are assumed to take values in negative fractional order spaces and may exhibit temporal singularities at the initial time. Our main results establish well-posedness and pathwise spatio-temporal regularity of the solutions when these coefficients are globally Lipschitz, and we prove the existence, uniqueness, and pathwise regularity of maximal local solutions when the coefficients are locally Lipschitz. The local Lipschitz condition is formulated with respect to a specific time-weighted norm. This enables the application of our theoretical results to linear stochastic partial differential equations, as well as to a broad class of models with non-globally Lipschitz nonlinearities. These include the stochastic Burgers, Allen–Cahn, Fisher–KPP, and Burgers–Fisher equations, alongside the Ginzburg–Landau system and models featuring exponential growth, such as the stochastic solid fuel ignition model.

We study the existence of equilibria for state constrained, convex generalized Nash equilibrium problems (GNEPs) coupled with hyperbolic partial differential equations (PDEs). Analogous problems have been addressed for state constrained GNEPs coupled with elliptic and parabolic PDEs, respectively, but the problematic regularity of the set-valued constraint maps has been a barrier for development of an existence theory in the hyperbolic case. This is mainly due to compactness issues with the strategy-to-state maps. Beyond existence, we also provide first-order optimality conditions for a class of fairly general linear hyperbolic PDEs and show its relevance for applications such as the wave equation, advertising dynamics, and the linearized isothermal Euler system on networks.

Weighted integrated observability inequalities for heat equations usually involve a small-time factor of the form $e^{-\gamma/t}$. We prove that this scale is not an artefact of Carleman or spectral methods: it is forced by the geometry of the observation set. Let $A$ be a nonnegative self-adjoint operator on sections of a finite-rank Euclidean vector bundle over a doubling metric measure space, satisfying ultracontractivity, Davies-Gaffney estimates (equivalently, finite speed of propagation for the wave equation) and a pointwise local Weyl law. If a weighted integrated observability inequality holds on a measurable set $\omega$, for a fixed horizon $T\in(0,+\infty]$ and an admissible weight $h$, then, for every $0<\kappa<\frac{1}{2}$, $$ h(t)\leq A_{T,\kappa}\exp\left(-\kappa\frac{\mathcal{L}(\omega)^2}{t}\right),\qquad 0<t<T, $$ where $\mathcal{L}(\omega)$ is the essential maximal distance to $\omega$, replaced by any finite radius when $\mathcal{L}(\omega)=+\infty$. Thus, for $h(t)=e^{-\gamma/t}$, necessarily $\gamma\geq\mathcal{L}(\omega)^2/2$. This settles, with the optimal threshold, the maximal-distance lower bound for the infinite-time constant left open in earlier work. In the control-norm convention, the fast-control rate is at least $\mathcal{L}(\omega)^2/4$, recovering Miller's bound. The proof rests on the spectral packet $(\cosh(r\sqrt A)-1)e^{-tA}$. A pointwise Weyl law gives its sharp lower growth, while finite propagation speed and a weak-kernel Kannai transmutation formula make it exponentially small on $\omega$. Without kernel continuity or compact resolvent, we develop pointwise spectral measures and weak wave kernels. The framework covers Laplace-type operators on compact Riemannian manifolds, coupled heat systems, Schrödinger operators on $\mathbb{R}^d$, equiregular sub-Laplacians and Grushin models, and $\delta'$-coupled Laplacians on metric graphs.

This article proves norm inflation in the critical Sobolev space $H^{5/2}(\mathbb{R})$ for the Fokas-Olver-Rosenau-Qiao equation, which is a modified Camassa-Holm-type equation with cubic nonlinearity. This result complements the well-posedness theory for this equation, which was previously known to be locally well-posed in $H^{s}(\mathbb{R})$ for $s>5/2$. The proof relies on the construction of explicit initial data satisfying the previously known blow-up criteria for the Fokas-Olver-Rosenau-Qiao equation, a step that appears to be of independent interest.

We introduce and analyze Darcy flow through a saturated porous medium subject to bilateral constraints on the normal flux across a portion of the boundary. The problem is posed as the maximization of a velocity-based dual concave energy over a convex subset of $H(\mathrm{div};\Omega)$; Fenchel duality identifies a pressure-based predual formulation, yields strong duality, and provides convex optimality conditions with a complementarity structure on the constrained boundary. The primal–dual gap satisfies an a posteriori error identity, free of generic constants, valid for arbitrary admissible approximations. The duality structure is inherited by a Raviart–Thomas/Crouzeix–Raviart discretization, from which we derive a discrete error identity and a priori error decay rates under fractional regularity assumptions on the solution and the flux bounds. Numerical experiments, including adaptive refinement driven by localized primal–dual gap indicators, support the theory.

We prove that maximizers exist and that maximizing sequences possess subsequences that converge modulo symmetries to maximizers for the $L^p \to L^q$ Fourier extension inequalities associated to the hyperbolic paraboloid in three ambient dimensions.

We consider the solvable behaviour for the initial value problem of time nonlocal evolution equations with the kernels of type $(\mathcal{PC})$. Our aim is to analyze some sufficient conditions ensuring local existence, integrability of mild solutions when the nonlinear term exhibits rapid growth. Moreover, a sufficient condition of the unsolvable result is also established. It turns out that the solvable behaviour is closely connected with the index on the initial value, which occurs a critical dimension phenomenon. The proofs rely on subordination and monotone iterative method. Finally, several examples are given to illustrate the wide applicability of the results.

In this note we focus our attention on a class of nonlocal-in-time evolution equations with Lévy diffusion, they arise as models of unidirectional viscoelastic fluid flow and physical phenomena with memory effect.We first consider the existence of the classical solution to a nonlocal linear evolution problem under conditions on the involved memory kernels which allows complete positivity. Then we investigate the limit of this model to a generalized Rayleigh-Stokes equation, as the index of Lévy diffusion gets concentrated near two, we prove that the solution of nonlocal-in-time problem with Lévy diffusion uniformly converges to that of the generalized Rayleigh-Stokes equation and reveal the convergence rate.Finally, the existence and limiting behavior of the mild solution to a nonlocal evolution problem with nonlinearity are established. The proofs are based on subordination principle and relaxation function theory.

We study the nonlocal $\rho$-Laplacian, defined as the composition of the nonlocal divergence and gradient operators associated with a general radial kernel $\rho$: $\Delta_\rho u=\mbox{div}_\rho\left(D_\rho u\right)$. Our first main contribution is to establish a precise connection between this operator and the class of integro-differential elliptic operators studied by Fernández-Real and Ros-Oton (\cite{FernandezRos}), identifying explicit conditions on the kernel $\rho$ that guarantee membership in this class. Our second main contribution concerns maximum and comparison principles for the $\rho$-Laplacian. We establish both a strong and a weak maximum principle under conditions on $\rho$ that are strictly weaker than those required for membership in the integro-differential class, thereby covering a genuinely broader family of operators. The results require only minimal assumptions on the kernel, and in particular do not rely on any fractional-type comparability condition.

One might expect that finite relaxed elastic energy rules out diffuse singularities in the derivative, leaving only absolutely continuous and jump parts. This is suggested by the role of $SBV$ in free-discontinuity problems and by interpreting jumps as vertical segments of limiting graphs. We show that it fails for the relaxed one-dimensional Willmore energy. We construct a continuous function $u\in BV((0,1))$ with $D^c u\neq0$ and $\overline{\mathcal{W}}(u)<\infty $, so finite relaxed Willmore energy does not imply $u\in SBV((0,1))$. The idea is to concentrate the Cantor part exactly where the absolutely continuous slope blows up. There the singular diffuse measure meets the blow-up condition of the relaxation theorem, while the weighted curvature term stays integrable. Geometrically, the example shows that Cantor parts of $BV$-graph derivatives need not be intrinsic singularities of the underlying curve. The graph has an arc-length parametrization of class $C^1\cap W^{2,2}$, and a suitable rotation turns it into a Lipschitz graph whose derivative has no singular part. The construction also rescales to make the relaxed Willmore energy arbitrarily small, and it extends to relaxed $L^p$-curvature energies for all $p>1$.

A chemotaxis-driven multiphase multispecies diffusion system, arising in the formation of vascular-like structures, is analyzed. The model couples porous-medium-type cross-diffusion equations for the volume fractions of the cellular components with multispecies Keller-Segel equations governing the chemoattractant concentrations, posed in a bounded domain with no-flux boundary conditions. The system is derived within a multiphase framework based on mass and force balance laws, together with a characterization of the mixture pressure gradient, which follows from the volume-filling constraint. The existence of a global weak solution, the weak-strong uniqueness property, the exponential decay to the constant steady state, and the vanishing diffusion limit are established. The existence proof extends the boundedness-by-entropy method to solution codomains that are bounded in some directions only, while the other results are based on various entropy estimates and uniform dissipation bounds.

We study diffusion equations with completely monotone memory when the instantaneous diffusion form is merely non-negative and may therefore lose coercivity. For a kernel whose Bernstein representing measure has finite total mass $M_{0}=\nu([0,\infty))$, we introduce an extended state consisting of the physical variable and its continuum of internal variables. The aggregation and constant-embedding operators are adjoint with respect to the memory energy, and the resulting cross-term cancellation makes the augmented generator $m$-dissipative. This yields a unique mild solution, Lipschitz dependence on the data, and a contraction estimate that contains no positive lower bound for the instantaneous form. The zero-prehistory trajectories form a memory graph space, in which the problem is well posed in the sense of Hadamard. If, in addition, the first Bernstein moment $M_{1}=\int_{[0,\infty)}\lambda\,\diff\nu(\lambda)$ is finite, the memory potential and first-moment field possess the regularity needed to identify the semigroup solution with an encoded weak formulation and to obtain explicit stability bounds. We further prove uniform norm-resolvent convergence and convergence of the associated semigroups when a coercive instantaneous contribution vanishes. Under an additional $L^{2}(0,\Tend;V)$-regularity assumption on the limiting solution, the convergence rate in the memory graph norm is $O(\varepsilon^{1/2})$. These results provide a continuous stability target for structure-preserving and certified discretisations of memory-dominated diffusion.

In spatially structured populations, rare neutral mutations can spread through large regions during a range expansion, a phenomenon known as gene surfing. Whether deleterious mutations can also surf remains poorly understood. To address this question, we study a deterministic version of the spatial Muller's ratchet, given by an infinite system of reaction-diffusion equations describing an asexual population subject to mutation, migration, and density-dependent reproduction and death. After establishing that the system of PDEs is well-posed, we analyse the distribution of deleterious mutations within the population. In the monostable regime, we derive quantitative bounds on the ratio between the density of individuals carrying a given number of mutations and the density of mutation-free individuals. Under a Fisher-KPP condition, we further determine the spreading speed of the population into an empty habitat, confirming non-rigorous computations of Foutel-Rodier and Etheridge. Finally, using a tracer dynamics approach, we show that deleterious mutations cannot surf deterministic waves: although they are present at the expansion front, they only arise as recent descendants of the wild type.

We show that for Schrödinger operators in a connected domain, the Hohenberg-Kohn theorem holds within the class of Laplace form-bounded external potentials if and only if the single-particle density is strictly positive quasi-everywhere. Furthermore, we show that this condition is satisfied for non-interacting Schrödinger operators whenever a ground-state exists. Consequently, we establish the Hohenberg-Kohn theorem for non-interacting systems, and thereby the uniqueness of the Kohn-Sham potential, within the maximal class of Laplace form-bounded distributions. The main ingredient to establish these results is a characterization of regular states, whose proof relies on tools from classical potential theory. Moreover, this characterization reveals that, in the continuum setting, the fundamental mechanism underlying the Hohenberg-Kohn theorem is the (quasi)-unique continuation of the density rather than of the many-body wavefunction.