Papers

In this paper, we consider the existence and stability of waves progressing through a finite-depth two-dimensional body of water that carry a vortex in their bulk. The waves are acted upon by gravity, and sit below a region of air at constant pressure, with the air–water interface being a free boundary along which capillary effects are incorporated. We consider two classes of vortices: point vortices, where formally the vorticity is a Dirac $\delta$, and hollow vortices, where the vortex core is a region of constant pressure outside the fluid domain and about which there is a nonzero circulation. First, we prove that steady small-amplitude wave-borne point vortices exist for any subcritical Froude number and supercritical Bond number, and then show they are conditionally orbitally stable. Second, through a vortex desingularization argument, we construct capillary-gravity wave-borne hollow vortices. Notably, the wave speed is $O(1)$ for both these families.

We prove the existence of solutions to the stochastic magnetohydrodynamics (MHD) system, where randomness is introduced through random initial data and a stochastic integral appearing solely in the induction equation, while the fluid equations remain deterministic. We focus on the case where the adiabatic exponent $\gamma$ satisfies $\gamma > \frac{3}{2}$. The existence proof is carried out using the penalization method. We define the notion of a martingale solution and establish sufficient conditions for its existence. The proof then proceeds by means of the stochastic compactness method. Using an energy inequality, we derive a priori estimates in terms of expectation. Due to the stochastic nature of the problem, we demonstrate convergence in law of the penalized solutions and verify that the limiting object is a martingale solution.

We set up and investigate an approximation scheme for a variational theory of cohesive fracture in antiplane shear deformations. We consider a family of functionals acting on scalar valued functions in general dimensions depending on a small parameter $0 < \varepsilon \ll 1$ and on two fields: the elastic part of the displacement field and an eigendeformation field that describes the inelastic response of the material beyond the elastic regime. We measure the inelastic contributions of the latter in terms of a non-local energy functional. Our main result shows that, as $\varepsilon \to 0$, the approximate functionals $\Gamma$-converge to a cohesive zone model.

We study the $L^p$-boundedness of Isozaki–Kitada type modified wave operators associated with Schrödinger operators $P=-\partial_x^2 + V$ in one dimension for potentials $V$ including long-range potentials such as negative Coulomb-like ones $V(x) = -(1+|x|^2)^{-\mu/2}$ for $\mu>0$. For such potentials, if $ \mu \in (1, 2)$, we prove that the middle and the high energy parts are bounded on $L^p (\mathbb{R})$ for any $p \in [1, \infty]$ but the low energy part is unbounded except for $p=2$. Moreover, if $\mu \in (0, 1]$, we show that even the middle energy part is unbounded except for $p=2$. Actually, regarding the $L^p$-boundedness in the middle and the high energy regimes, we give a complete classification of slowly decaying potentials without imposing any sign condition.

Nitsche proved that every smooth Jordan curve in $\mathbb{R}^3$ of total curvature at most $4\pi$ bounds a unique minimal disk, which is moreover strictly stable. We prove an analogue of this result for Riemannian $3$-balls with mean convex boundary, under an explicit pinching condition on the negative sectional curvature, together with a bound on the covariant derivative of the Ricci tensor. In this setting, every smooth Jordan curve in the boundary sphere of total curvature at most $4\pi$ bounds a unique embedded minimal disk which is strictly stable.

The present paper studies integral geometry problems on three-dimensional Riemannian balls, where integration is performed over minimal surfaces or, more generally, $\Phi$-surfaces defined by an elliptic curvature functional $\Phi$. The space of all $\Phi$-surfaces spanned by round circles on the boundary is a three-dimensional manifold, which we call the space of circles. We show that, when the metric is $\Phi$-simple - a notion which extends to this setting the notion of simple metrics in the geodesic case -, the Gauss lifts of the $\Phi$-surfaces define a foliation of the unit tangent bundle that should be viewed as a two-dimensional analogue of the standard geodesic foliation. This is achieved by solving a foliated Plateau problem on the ball. We then analyze the associated surface Radon transform corresponding to integration along the surfaces and show that it has a finite-dimensional kernel; we also prove that it is injective for an open and dense set of metrics. In the special case of a foliation by minimal surfaces, we apply these results to solve the following boundary area rigidity problem: does the collection of areas of the minimal surfaces determine the metric up to isometry?

The bifurcation phenomenon in a singularly perturbed one-phase free boundary problem governed by the g-Laplacian is established under prescribed boundary conditions. This bifurcation is characterized by the existence of a third solution, obtained via the Mountain Pass Lemma, when the boundary data decreases below a suitable threshold. Moreover, we analyze the asymptotic behavior of the associated evolution problem, proving convergence to stable stationary solutions and showing that the Mountain Pass solution is unstable in this dynamical sense.

We establish intrinsic continuation criteria and finite-energy global solvability for the stochastic tamed Navier–Stokes equations on $\mathbb R^3$ driven simultaneously by multiplicative cylindrical Wiener and compensated Poisson noise. Under local coefficient hypotheses, the maximal local $L^p$ solution, $p>3$, satisfies a blow-up alternative independent of auxiliary cutoffs and a Serrin criterion with time exponent $2p/(p-3)$. Additional coercivity of the taming term and compatible $L^2$ and gradient noise bounds yield the global well-posedness theory for divergence-free initial data $u_0\in L^p(\Omega;L^p)\cap L^2(\Omega;L^2)$. No smallness or initial $H^1$ regularity is required. The solution has càdlàg $L^p\cap L^2$ paths and gains $H^1$ regularity at positive times, with time-weighted $H^1$ and $H^2$ estimates. The proof combines finite-energy persistence with endpoint completion that retains terminal Poisson jumps and permits restart in the original uniqueness class.

We study the tangent bundle $T_\mu$ to a Radon measure $\mu$ in Euclidean space, introduced by Bouchitté, Champion and Jimenez (2005). Its construction, inspired by Monge-Kantorovich optimal transport theory, involves the duality between Lipschitz functions and the so-called Arens-Eells space, a Banach subspace of distributions obtained by completing the set of balanced signed measures. Precisely, a velocity field $\sigma$ is $\mu$-tangent when the divergence of $\sigma \mu$ lies in the Arens-Eells space. The tangent bundle $T_\mu$ defined $\mu$-almost everywhere provides a local projection that allows to construct a $\mu$-tangential gradient operator on Lipschitz functions that is weakly continuous and enjoys integration by parts. In this paper, we introduce a new quantitative estimate involving the tangential and normal components of a given velocity vector field $\sigma \in L^1_\mu(\mathbb{R}^d)$. Specifically, we show that the tangential condition $\sigma \in T_\mu$ holding $\mu$ a.e. is equivalent to each of the following two conditions: the convergence in the Arens-Eells space of $h^{-1} \left((id+h \sigma)_{\#} \mu - \mu\right)$ as $h \to 0$, and the differentiability of Lipschitz functions along $\sigma$. Moreover, given a field of non-tangent directions, we construct a Lipschitz function that is not differentiable on a large set, confirming that $T_{\mu}$ agrees with the decomposability bundle of Alberti and Marchese. We finally relate $T_\mu$ to the tangent measures of Preiss and survey further properties of the tangential differential calculus.

Let $\Omega$ be an open bounded connected subset of $\mathbb{R}^N$, $N \geq 2$, of class $C^{2,\alpha}$, for $\alpha \in (0,1)$. Let $p \geq 2$ and $\beta>0$. We prove the symmetry of the solution of the $p$-torsion problem with Robin boundary condition subject to the natural overdetermined condition coming from a shape derivative argument and to an extra condition on $\beta$ and on the minimum of the principal curvatures of $\partial\Omega$. The proof is based on some new integral identities, involving the linearized operator of the $p$-Laplacian applied to the standard $P$-function. In passing we prove some other rigidity results in the spirit of Serrin's and Alexandrov's Theorems.

We consider a hierarchy of models that can be obtained from the original Nyzhnyk system by imposing conditions on parameters, introducing potentials or pseudopotentials, performing limiting processes with respect to a scaling parameter, applying differential substitutions and interpreting parts of independent and/or dependent variables as complex or real. Therefore, among the Nyzhnyk models, one can distinguish between symmetric and asymmetric, dispersive and dispersionless, standard and modified, as well as real, complex, mixed and specific models. One can also distinguish single partial differential equations or systems of such equations, as well as linear or nonlinear Lax representations in the dispersive or dispersionless cases, respectively. For each specified model, we compute the maximal Lie invariance pseudoalgebra and, in the symmetric case, find the point- and contact-symmetry pseudogroups using the megaideal-based version of the algebraic method. It is shown that relations between these pseudoalgebras and between these pseudogroups are induced by relations between the corresponding models. Defining (finite-dimensional) subalgebras are singled out in all these pseudoalgebras in the symmetric and specific cases. Based on the established correspondences between the dispersionless and dispersive models, we completely classify one- and two-dimensional subalgebras of the maximal Lie invariance pseudoalgebra of the (dispersive symmetric potential) Nyzhnyk equation and one-dimensional subalgebras of the maximal Lie invariance pseudoalgebra of its linear Lax representation.

In this paper, we study a two-dimensional Poisson–Nernst–Planck–Navier–Stokes system in a periodic channel driven by wall motion and an imposed tangential electric field, with two ionic species that may have unequal diffusivities. At the electroneutral Couette state, a diffusivity-weighted ionic energy combines with the triangular structure of the linearized operator to yield exponential linear stability for fixed \((A,E_0)\) and all positive diffusivities. Small-data nonlinear exponential stability is then obtained on a complex interpolation space adapted to the graph domain of the generator by combining analytic-semigroup smoothing with quadratic estimates. For unequal diffusivities, the nonzero streamwise modes of the linearized ionic subsystem satisfy a bounded-channel enhanced-dissipation estimate at rate \(D_{\min}^{1/3}|A|^{2/3}\) under an explicit strong-shear condition. When \(D_+=D_-\), a Fourier-mode factorization separates the common Couette advection–diffusion operator from a contractive drift–reaction semigroup, so the same scalar mixing rate is retained without any smallness condition on the electrostatic coupling. We also construct an exact Poisson–Boltzmann/electroosmotic Couette family for prescribed wall potentials. For sufficiently small wall-potential amplitude, its linearized generator is a small graph-domain perturbation of the electroneutral generator, which yields linear and nonlinear exponential stability on the same interpolation scale.

Let $\Omega\subset\mathbb{R}^N$ be a bounded convex domain, and let $A=-\Delta_D$ be the Dirichlet Laplacian. For every $s>0$ with $N>2s$, we prove that the Green-kernel transform $(x,y)\mapsto K_{s,\Omega}(x,y)^{-1/(N-2s)}$ extends continuously by zero to the diagonal and is jointly convex on $\Omega\times\Omega$. We prove that the regular part extends real analytically across the diagonal and that the associated Robin function is strictly convex and diverges to $+\infty$ at the boundary. Hence it has a unique critical point. If $\Omega$ is of class $C^{2,\vartheta}$, this point is nondegenerate for every $s>0$ with $N>2s$. Moreover, the Hessian of the Robin function is positive definite throughout $\Omega$ when $0<s<1$ or $N>3s$. The arguments combine convexity inequalities for Dirichlet heat kernels, a weighted boundary identity at fractional orders, and a finite-part transfer of Hessian positivity between spectral powers. In particular, the results apply to spectral fractional Dirichlet Laplacians and Navier polyharmonic operators.

We establish a pointwise lower bound for the heat kernel of the elliptic operator $ \Lambda=(1+|x|^\alpha)\Delta +b|x|^{\alpha-2}x\cdot\nabla-|x|^\beta$, where $d\geq3$, $\alpha>2$, $\beta>\alpha-2$, and $b\in\mathbb R$. For every $\tau>0$, we prove that $$ \begin{aligned} p(t,x,y) &\geq C_\tau e^{\lambda_0t} \left(\frac{1+|y|^\alpha}{1+|x|^\alpha}\right)^{\frac{b}{2\alpha}} \frac{(|x||y|)^{-\frac{d-1}{2}-\frac{\beta-\alpha}{4}}}{1+|y|^\alpha}\\ &\quad\times\exp \left[ -\int_1^{|x|}\sqrt{\frac{s^\beta}{1+s^\alpha}}\,\mathrm ds -\int_1^{|y|}\sqrt{\frac{s^\beta}{1+s^\alpha}}\,\mathrm ds \right] \end{aligned} $$ for all $t\geq\tau$ and $|x|,|y|\geq1$, where $\lambda_0<0$ is the largest eigenvalue of $\Lambda$ and $C_\tau>0$ is independent of $t,x,y$. The proof applies the classical Davies - Simon argument in a weighted symmetric setting.

We derive an explicit Faber-Walsh representation for the planar conductivity problem with finitely many disjoint inclusions of general shape and independently chosen positive conductivities. Walsh's lemniscatic map gives a conformal coordinate without smallness or wide-separation assumptions. The field expansion is expressed through Faber-Walsh polarization tensors, which we relate to the complex contracted generalized polarization tensors by a triangular change of basis. For analytic interfaces, we prove conformal continuation and absolute and uniform convergence of the Walsh-Grunsky series near the boundary to obtain explicit factorizations of the Neumann-Poincaré operator and the Faber-Walsh polarization tensors. These factorizations separate canonical interaction, which can remain nonzero even when all Grunsky coefficients vanish, from conformal deformation, while retaining the componentwise conductivity dependence in a coupled resolvent. For harmonic polynomial incident fields, we prove geometric convergence of outgoing truncations with exact coefficients on compact subsets of fixed exterior branch domains. Numerical comparisons with independent boundary-integral solutions illustrate the representation for heterogeneous, asymmetric, and nearly touching configurations.

This paper investigates entire classical separable variable radial solutions to a class of parabolic k-Hessian equations -u_t (sigma_k(lambda(D^2u)))^alpha = f(|x|)g(t), where the right-hand side consists of positive continuous functions. Without assuming that -u_t is positively bounded, we extend previous work for the equations whose right-hand side is 1 to more general cases, including cases where the right-hand side terms are bounded and periodic. By employing the method of Euler's broken line, we obtain the existence and nonexistence results. Furthermore, we obtain the precise asymptotic power exponent.

Let H^{n+1} be hyperbolic space of sectional curvature -1, with a fixed point o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^{alpha} {nu} of smooth, closed, strictly h-convex hypersurfaces enclosing o, where r is the geodesic distance to o and alpha>0. The radial weight f is modeled on sinh^{beta}r; we treat both regimes: beta>1+k{alpha} and beta=1+k{alpha}. Under a structural condition of f, the flow exists smoothly for all time, preserves strict h-convexity, and contracts to o. The normalized radial function converges, exponentially in normalized time: to 1 and to a positive constant R_{infty} respectively in two different cases.

In this paper, we investigate the compactness problem for the super-Liouville equation under Dirichlet-chiral boundary conditions. We prove that no blow-up occurs in this setting.

By using spherical harmonicas decomposition and Gaussian-type Poincaré inequalities, we establish several sharp stability estimates for the following second-order weighted Heisenberg Uncertainty Principle \begin{equation*} \int_{\mathbb{R}^N} \!\frac{|\Delta u|^2} {|x|^{2\alpha}} \mathrm{d}x \int_{\mathbb{R}^N} |x|^{2\alpha+2}|\nabla u|^2\mathrm{d}x \geq \frac{(N+4\alpha+2)^2}{4} \left(\int_{\mathbb{R}^N} |\nabla u|^2 \mathrm{d}x\right)^2, \end{equation*} and \begin{equation*} \int_{\mathbb{R}^{N}} \frac{|\Delta u|^{2}} {|x|^{2\alpha}} \mathrm{d}x +\int_{\mathbb{R}^{N}} \left|x\right|^{2\alpha+2} |\nabla u|^{2}\mathrm{d}x \ge\left(N+4\alpha+2\right) \int_{\mathbb{R}^{N}} |\nabla u|^{2}\mathrm{d}x. \end{equation*} We also provide the explicit value and the necessary and sufficient condition for attainability of the sharp stability constants. Moreover, when $\alpha=0$, our results reduce into those of [Calc. Var. Partial Differential Equations 64 (2025), Paper No. 129], [J. Funct. Anal. 290 (2026), Paper No. 111321] and [arXiv:2510.00453].

We prove that, for vector-valued convolution Calderón–Zygmund operators, boundedness on a single nontrivial Morrey space is equivalent to the corresponding global $L^p$ boundedness. Thus one Morrey scale already contains the full finite-$p$ boundedness information. The implication from Morrey to $L^p$ is obtained by a separated-copy amplification argument that reconstructs the global norm from a single scale-local estimate; the converse is proved in the same vector-valued framework by a local/far-field decomposition. As a consequence, boundedness of the vector-valued Hilbert transform on one nontrivial Morrey space is equivalent to the UMD property. The result shows that Morrey estimates do not bypass the classical Banach-space obstruction: they detect it exactly.

We develop compactness and defect-of-compactness results for Banach-valued evolution classes with Morrey control in time. For \[ \begin{aligned} \mathbb W_M^{p,\lambda}(0,T;E_0,E_1) =\{u\in\mathcal M^{p,\lambda}(0,T;E_0):\;& u'\in\mathcal M^{p,\lambda}(0,T;E_1)\},\\[-1mm] &0<\lambda<1. \end{aligned} \] the exact trace exponent \[ \theta=1-\frac{1-\lambda}{p} \] governs both continuity and compactness. If $E_0\hookrightarrow\!\hookrightarrow E\hookrightarrow E_1$, bounded sets are compact in the same Morrey space $\mathcal M^{p,\lambda}(0,T;E)$, not merely in $L^p(0,T;E)$. In contrast, compactness in $C([0,T];E)$ holds if and only if the exact trace space $(E_1,E_0)_{\theta,\infty}$ embeds compactly into $E$. We also obtain compact lower-order Hölder embeddings and a sharp Hilbert-triple threshold. On unbounded domains we prove a tightness criterion for global Morrey compactness. In the Hilbert-valued case we establish, by a direct time-averaging argument, cocompactness modulo spatial translations and a translation profile decomposition whose remainder vanishes in every strictly subcritical time-Morrey–Sobolev target; a uniform little-Morrey condition removes the loss in the time exponent. At the doubly critical endpoint, heat and whole-space Stokes dynamics force balanced parabolic profiles, and the remainder vanishes in $\mathcal M_t^{2,\lambda}L_x^{2^*}$. Finally, in three-dimensional Navier–Stokes we show that the classical nonlinear profile decomposition has a scale-sensitive Morrey refinement: the remainder is small in $\mathcal M_t^{p,1-p/4}L_x^6$ for every $2\le p<4$, and orthogonal profiles have vanishing interaction in the corresponding Morrey forcing space.

Trace spaces determine exactly which initial values are compatible with an evolution class. We identify the exact trace generated by vector-valued Morrey control in time and show that it differs essentially from the classical $L^p$ theory. For a Banach couple $X_1\hookrightarrow X_0$, the trace of the natural Morrey evolution class is the weak real-interpolation space $(X_0,X_1)_{\theta,\infty}$, where $\theta=1-(1-\lambda)/p$. Every element of this space occurs as a trace through a bounded extension operator. The result is sharp in both interpolation parameters: in general the smoothness exponent cannot be increased and the fine index $\infty$ cannot be replaced by any finite index. Thus Morrey control changes the exact trace mechanism rather than merely strengthening an integrability estimate. At the limiting endpoint, bounded mean oscillation (BMO) control yields finite-index interpolation traces together with a logarithmic modulus of continuity. The result isolates the precise initial-data space naturally associated with local, scale-sensitive time regularity and provides a trace framework suited to evolution equations with Morrey-type maximal regularity. \keywords{Morrey spaces \and trace spaces \and real interpolation \and evolution equations \and bounded mean oscillation}

We prove global well-posedness of the one-dimensional defocusing cubic nonlinear Schrödinger equation in the modulation space $M^{\infty,1}(\mathbb{R})$. This space imposes no spatial decay and contains $C_b^2(\mathbb{R})$ as well as all absolutely convergent sums of plane waves. The result applies to arbitrary data in this space, including large smooth quasiperiodic profiles and their localized perturbations. The proof constructs a nonnegative density satisfying a local conservation law from forward Weyl ratios, which are defined through half-line square-integrable solutions of the associated spectral problem. A suitable nonlinear combination of localized integrals of this density controls the modulation norm. Finally, choosing the spatial localization scale and NLS scaling in a coordinated way makes the accumulated boundary flux small enough to continue every mild solution globally.

This paper investigates the following May–Nowak type model for viral infection in a bounded domain $\Omega \subset \mathbb{R}^n$, $n \ge 2$: $\begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v) + \kappa - u - uw - \mu \dfrac{u^{1+\alpha}}{\ln^k(u+e)}, \\[1mm] \gamma v_t = \Delta v - v + uw, \\[1mm] w_t = \Delta w - w + v, \end{cases}$ where $\chi \in \mathbb{R}$, $\mu > 0$, $k \in [0,1)$, $\alpha > 0$, and $\gamma \in \{0,1\}$. We establish the global existence and uniform-in-time boundedness of classical solutions for suitably regular initial data under one of the following conditions: (A) $\gamma = 1$, $n = 2$, $k \in [0,1)$, $\alpha = 1$, and $\mu > 0$; (B) $\gamma = 1$, $3 \le n \le 5$, $k = 0$, $\alpha = 1$, and $\mu $ is sufficiently large; (C) $\gamma = 1$, $n \ge 6$, $k = 0$, $\alpha > \frac{n-2}{4}$, and $\mu > 0$; (D) $\gamma = 0$, $n \ge 2$, $k = 0$, $\alpha > \frac{n+2}{2}$, and $\mu>0$. In particular, in the physically relevant dimensions $n = 2,3$, both subquadratic and quadratic damping are sufficient to prevent blow-up. Moreover, in the fully parabolic case ($\gamma = 1$), our results improve upon recent findings by relaxing the condition $\alpha > \frac{n}{2}$ to weaker assumptions within the above parameter regimes.

For every $n\ge2$ and $1<p<\infty$, we show an existence of a nonconstant homogeneous entire $p$-harmonic function whose only critical point is the origin. The angular parts of these functions are axially symmetric and even across the equator. To establish the solution we use contraction argument at the pole in conjuction with a shooting argument. It follows that in every dimension $n\ge3$ and for $p<2$ sufficiently close to $2$, the natural gradient has pointwise Hölder exponent strictly below one at the origin. Such examples settle the remaining scalar case of the conjecture of Balci, Diening, and Weimar by disproving both the $C^1$ assertion and the linear $L^2$ mean oscillation estimate.

The main objective of this paper is to investigate orthonormal Strichartz estimates for Schrödinger equation on the Wiener amalgam space $\mathcal{W}(\mathcal{F} L^p, L^q)$. More precisely, we first examine improvements in the time-integrability exponent of the existing Strichartz estimates established by Cordero and Nicola in \cite{NFC} for Schrödinger equations associated with $\mathcal{H}^{+}=-\frac{1}{4\pi}\Delta+|x|^2$. We then extend these improved Strichartz estimates from a single initial datum to systems of orthonormal families, providing, to the best of our knowledge, the first results in this direction in the setting of Wiener amalgam spaces. We further extend the classical Strichartz estimates in Wiener amalgam spaces from a single initial datum to systems of orthonormal families of initial data for the Schrödinger equation associated with the operator $\mathcal{H}^{-}=-\frac{1}{4\pi}\Delta-|x|^2$ and Hamiltonian operator of the form $\mathcal{H}_{\mathcal{A}} = -\frac{1}{4\pi} B \nabla \cdot \nabla$, where $\mathcal{A} = \begin{pmatrix} 0 & B \\ 0 & 0 \end{pmatrix} \in \mathrm{Sp}(d,\mathbb{R})$ with $B = B^*$ and $\det B \neq0$. A key ingredient of our approach is Stein's complex interpolation theory for Wiener amalgam spaces, combined with a duality argument inspired by the work of Frank and Sabin. As an application of these orthonormal estimates associated with $\mathcal{H}^{+}, \mathcal{H}^{-}$, and $H_{\mathcal{A}},$ we establish local and small-data global well-posedness for the Hartree equation with infinitely many particles, for non-trace-class initial data.

We disprove almost everywhere convergence of the Schrödinger evolution on the standard torus \(\mathbb{T}^d\) for initial data \(f\in H^s(\mathbb{T}^d)\) when \(s<d/(d+2)\), in all dimensions \(d\ge2\). We construct normalized data with frequencies of size $N$ whose evolution attains size $N^{d/(d+2)}$ on a set of uniformly positive measure. The key ingredient in the construction is that the frequencies lie in an affine congruence class, so that at suitable rational times their phases coincide on a family of well-separated spatial points. The result matches the known estimate for $s>d/(d+2)$ with $d\geq2$. By integer dilation and uniform boundedness, we also obtain a single datum in $H^s$ whose evolution is unbounded along a sequence of times tending to zero whenever $s<d/(d+2)$. We also record a logarithmic upper bound in $2D$ at the critical frequency power and a lower bound on shrinking time intervals.

A multispecies, collisionless plasma is modeled by the relativistic Vlasov-Maxwell system. Assuming the plasma is globally neutral, we show that solutions can be constructed with arbitrarily fast, polynomial rates of decay, depending upon the cancellation properties of the limiting spatial averages, as well as the masses and charges of the particles. In doing so, we establish a countably infinite number of asymptotic profiles for the charge and current density, electric and magnetic fields, and their derivatives, each of which is necessarily realized by a sufficiently smooth solution. Our approach relies upon a refined field representation that features self-similar behavior away from the boundary of the light cone. In each case we establish a linear scattering result in $L^\infty$ for every particle distribution function, namely we show that they converge as $t \to \infty$ along the transported spatial characteristics at increasingly faster rates. In the case that charge cancellation does not occur, our methods can be further extended to demonstrate polyhomogeneous expansions with logarithmic corrections.

We study weak solutions to degenerate parabolic double phase equations with growth exponents $2\le p<q$ and a modulating coefficient that is Hölder continuous with exponent $\alpha$. If the solution itself is Hölder continuous with exponent $\gamma$, we prove higher integrability of its gradient under the gap condition $q\le p+\alpha/(1-\gamma)$ together with $q<p+1$. This is the first gap bound of Hölder interpolative type for parabolic double phase problems, and it is the parabolic counterpart of the corresponding bound for elliptic problems. For $\alpha<1$ it allows exponents beyond the range $q\le p+\alpha$ known for bounded solutions.

This paper investigates the null controllability of the complex cubic Ginzburg–Landau equation with controls supported on general space-time measurable sets. Starting from a positive-measure control set, we use a slicing argument to obtain time slices of uniformly positive spatial measure. This yields uniform spectral inequalities on the slices, which are the key ingredient in constructing quantitative stabilizing feedbacks. A density-point argument provides a sequence of shrinking intervals accumulating at the density point, on which the active times have a uniform positive density. We then apply a time-iteration scheme with piecewise feedback laws and increasing decay rates. Iterating the resulting stabilization estimates shows that the closed-loop solution reaches the zero state at the selected density point. This provides a constructive approach to the local null controllability on measurable control sets for nonlinear parabolic equations.