Papers

In recent years, transformer architectures have revolutionized the field of language processing, opening the door to previously unforeseen possibilities. However, from a theoretical point of view, the mathematical models proposed in the literature often lack direct contact with the actual architectures and depend on strong simplifying assumptions. In this paper, we reduce this gap by modelling the data flow in multi-headed transformer architectures as time-dependent gradient flows for a suitable interaction energy capturing the design of the attention mechanism. The explicit dependence on time allows us to consider different weights for each head and for each layer, without imposing constraints on the initialization method. Moreover, we prove that, under a suitable integrability assumption on the evolution of the weights, each element of the $\omega$-limit set of the gradient flows is a stationary point of the interaction energy at a limiting weight distribution. Finally, we analyse the stability of the gradient flows considering perturbations of both the initial data and the weights. Specifically, on the one hand, we study the robustness of the proposed models with respect to noisy inputs, establishing a continuous dependence of the gradient flows on the initial data and uniqueness of the flows. On the other hand, we prove the $\Gamma$-convergence of the perturbed interaction energy to the unperturbed one, leading to the convergence of the corresponding gradient flows. We complement these theoretical results with numerical experiments that confirm the predicted energy-dissipation identity and clarify the asymptotic behavior of the dynamics in both the autonomous-like (Ornstein–Uhlenbeck) and the genuinely non-autonomous (oscillating-weights) regimes.

We present a unified framework for the rigorous derivation of conservation laws and related identities for nonlinear Schrödinger equations with power-type nonlinearities. This approach treats the equation in its Duhamel form and uses the space-time integrability provided by Strichartz estimates, without relying on smooth approximations or regularization procedures. It was first introduced by the third author in [20] and subsequently developed in [7, 13]. In this paper, we establish a single integral identity from which all of the laws and identities considered here follow systematically. These include the conservation of charge (mass), energy, and momentum, the pseudo-conformal conservation law, and virial-type identities.

We study the homogenization of a class of non-local functionals featuring a rapidly oscillating periodic weight. By means of two-scale convergence, we explicitly evaluate the {\Gamma}-limit for constant target functions, revealing how the interplay between periodicity and non-locality forces the minimizing sequences to develop highly oscillating microstructures. As a natural consequence, we establish that the effective macroscopic functional fails to admit a standard double-integral representation.

An account is given on newest developments on $p$-adic boundary value problems on $p$-adic analytic manifolds and their relationship with diffusion. In particular, novel coordinate Laplacians on $p$-adic analytic $n$-manifolds constructed with the help of frame bundles, are introduced in this context. These are used to construct elliptic operators. Related Dirichlet problems are formulated and solved, generalising results on compact subdomains of $p$-adic $n$-space. In the end, an outlook towards number-theoretic applications as well as extensions of this theory to ultrametric analytic manifolds is given. This is a substantial upgrade of the presentation given at Branko's 80-th Birthday Conference in Belgrade, May 2025.

We consider the Cauchy problem for third-order evolution differential operators with variable coefficients, depending on time $t\in [0,T]$ and space $x\in\mathbb{R}$, where the leading coefficient $a_3(t)$ vanishes at $t = 0$ with finite order. We establish sufficient conditions on the behavior of the lower order coefficients $a_j(t,x)$ $j=1,2$ as $t \to 0^{+}$ and $|x| \to \infty$ that ensure well-posedness in $L^2(\mathbb{R})$, $H^{\infty}(\mathbb{R})$ and Gevrey-type spaces.

The compressible barotropic Navier–Stokes equations in vector-invariant form preserve the vorticity structure of the system and underlie modern atmospheric and ocean dynamical cores, yet no PDE theory has been developed for the compressible discrete system in this form. On a Delaunay–Voronoi mesh we prove via discrete exterior calculus, that every density-independent mass matrix with integration-by-parts-consistent divergence carries a sharp $\OO(h^2)$ energy residual of indeterminate sign that no operator choice can eliminate. This no-go theorem covers A-, B-, C-, D-, and quasi-B-grid staggerings. The density-weighted mass matrix is the unique algebraic remedy: it restores exact total energy while preserving the vector-invariant momentum equation, Lamb antisymmetry, and the topological conservation laws, at the cost of an $\OO(h^{r_\star})$ Kelvin defect matching the convergence rate. The residual is the cause of the Hollingsworth instability that has shaped vector-invariant dynamical-core design; the density-weighted construction removes it structurally. For the density-weighted~(DW) scheme on closed oriented Riemannian manifolds in $d = 2, 3$ we establish global well-posedness for $\nu \ge 0$, convergence to smooth solutions uniformly in $\nu$, and asymptotic preservation in the low-Mach limit; the density-free residual diverges as $\OO(M^{-1})$. Via a discrete Arnold energy-Casimir construction, exact discrete conservation forces Lyapunov stability around three classes of equilibria, excluding Hollingsworth instability: unconditional stability around hydrostatic and constant-flow stratified states, and conditional stability around sheared baroclinic states under a discrete Charney–Stern criterion. The DW scheme admits genuine baroclinic instability only when the continuum equations themselves do.

We study vortex stretching for the three-dimensional axisymmetric Euler equations without swirl in vorticity formulation. Danchin (2007) established global existence and uniqueness for bounded vorticity $\omega_0$ provided $\omega_0/r$ lies in the endpoint Lorentz space $L^{3,1}(\mathbb{R}^3)$ (together with a decay assumption on $\omega_0$). We prove that this $L^{3,1}$ endpoint is sharp: for every Lorentz exponent $q>1$, we construct multi-ring data $\omega_0 \in L^\infty (\mathbb{R}^3)$ with $\omega_0/r\in L^{3,q}(\mathbb{R}^3)$ that produce $L^\infty$-norm inflation of the vorticity; moreover, within the same class, we obtain instantaneous blow-up from data with infinitely many rings. Our initial data are inspired by the Kim–Jeong dyadic ring superposition (2022), but we crucially generalize it by allowing flexible conical support geometry for the ring profile. In the regime where outer rings are dominant – a multiscale viewpoint appearing in recent works including Kim–Jeong (2022) and Cordoba–Martinez-Zoroa–Zheng (2025) – we obtain a forward-in-time ODE cascade for ring amplitudes and aspect ratios in which vortex stretching weakens its own future forcing: as a ring amplifies, incompressibility flattens it, the aspect ratio collapses, and the induced stretching coefficient is geometrically depleted. A key new ingredient is a profile-localization argument that freezes the relevant Biot–Savart kernel and makes this depletion explicit, enabling us to exploit a monotone "productive window" (controlled by the cone slope) together with an exact cascade identity. This propagates stretching across scales and gives a robust lower bound on cumulative stretching, yielding ill-posedness in the full range $q>1$.

This is the first part of a series of papers proving the nonlinear stability of a one-parameter family of continuously self-similar $C^{1,\alpha}$ naked singularity solutions, with $0<\alpha\ll1$, to the spherically symmetric Einstein-scalar field equations. The stability holds for initial perturbations lying in a small open neighborhood of the data generating these naked singularity solutions, measured in a localized Hölder topology. These continuously self-similar naked singularity spacetimes were previously constructed by Christodoulou [D. Christodoulou, Examples of naked singularity formation in the gravitational collapse of a scalar field, Ann. of Math. 140 (1994), 607–653], who also proved their instability to black hole formation under sufficiently rough perturbations [D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. 149 (1999), 183–217], thereby verifying weak cosmic censorship within a rough functional framework. In complete contrast, in this paper, we obtain the first nonlinear stability of these naked singularity spacetimes under general perturbations of the same regularity as the background. We rely on the linearized stability result established in the companion paper [J. Singh and W. Zheng, Nonlinear stability of continuously self-similar naked singularities for the Einstein–scalar field equations II: linearized stability]. Our result underscores the decisive role of the functional framework in formulating the Weak Cosmic Censorship conjecture.

We consider 3D micropolar flows with possible vanishing spin viscosity and investigate the decay of the energy for large times. We compute first the exact $L^2$-asymptotic profile, as $t\to+\infty$, for solutions to the linear 3D micropolar equations, up to the second order. For the nonlinear micropolar system, we first establish the existence of restricted Leray solutions. This new notion of solutions is required because it is not known whether the weak finite energy solutions verify a strong energy inequality. Next, we study the large-time behavior of restricted Leray solutions, and prove that they behave asymptotically in $L^2$ like their linear counterpart, up to the critical algebraic decay rate $O(t^{-5/2})$ for the energy. Applying a remarkable linear enstrophy identity, we show that the microrotation field exhibits faster decay in $L^2$ than the velocity field, allowing us to impose our hypothesis on the velocity field only and not on the angular velocity.

We investigate the role of the four viscosity parameters, in fluids where the particles possess a microstructure (micropolar flows) and are allowed to rotate in a two-dimensional setting. We first establish the existence of global finite energy solutions, satisfying the classical energy equality, for arbitrary initial data in $L^2$, in the case of a spin viscosity $\gamma\ge0$, and we construct the asymptotic profiles of the solution as $t\to+\infty$. We deduce the remarkable fact that the large time behavior only depends on the kinematic viscosity $\mu$, and not on the other parameters $\chi$ (vortex-viscosity), $\gamma$ (spin viscosity) and $\kappa$ (gyroviscosity) of the model. Our primary tool is a new enstrophy-like identity of independent interest, involving the difference between the fluid vorticity and the micro-angular velocity. Another consequence of our analysis is the identification of scenarios where the presence of micro-rotational effects significantly enhances dissipation, thereby slowing down the fluid motion at large times.

This is the second part of a series of papers proving the nonlinear stability of a one-parameter family of continuous self-similar $C^{1,\alpha}$ naked singularity solutions, with $0<\alpha\ll1$, to the spherically symmetric Einstein-scalar field equations. These solutions were constructed by Christodoulou and are known to be unstable under sufficiently rough perturbations due to the blue-shift instability mechanism. In complete contrast to the previous instability results, we establish the linearized stability for those naked singularity spacetimes under perturbations of the same regularity as the background, revealing the central role of regularity in determining the strength of the blue-shift instability mechanism, and showing that it is not triggered at the regularity level of the background spacetime. The linear analysis carried out in this paper provides the foundation for the nonlinear stability result established in the companion paper [W. Zheng, Nonlinear stability of the continuous self-similar naked singularities for the Einstein-scalar field equations I: main results]. Together with that companion paper, this yields the nonlinear stability of these continuously self-similar naked singularities.

We study the onset of spatial instabilities in reaction networks where the spatially homogeneous system admits a steady state parameterization. We formulate a sufficient condition – based on the signs of the constant and leading coefficients of the characteristic polynomial of the linearized Jacobian scaled by the diffusion coefficients – that guarantees a Turing-like instability to spatially inhomogeneous solutions on appropriately chosen domains $\Omega$. We also present a specific condition on the domain size $|\Omega|$ required to trigger this instability. As a consequence of employing a monomial parameterization, these conditions take the form of algebraic polynomial inequalities involving only rate constants and diffusion coefficients. We apply these ideas to a network describing the sequential and distributive (de-)phosphorylation of a protein at two binding sites, ultimately deriving a condition involving only the four catalytic constants of the enzymes and the diffusion coefficients of the four enzyme-substrate complexes that guarantees a Turing-like instability.

We study the rate of convergence to the steady state in the True Moving Bed model of linear chromatography, as a function of the six parameters that appear in the model. The model is a system of eight linear partial differential equations of hyperbolic type, coupled through the equations themselves and also through boundary conditions. We prove that the rate of convergence is given by a dominant eigenvalue, whose existence we prove by means of the Krein-Rutman Theorem, and by comparison arguments. We show how to construct a (not at all simple) characteristic function, whose roots are the eigenvalues. We also study the asymptotic profile of the solutions for large times, although this part is not purely analytical, but a combination of analytical and numerical techniques. Beyond the theoretical results, these models also offer explicit quantitative information: we apply all our results to a Case Study, namely the separation of omeprazole enantiomers. Finally, we consider a simpler limit case, where all the calculations become explicit.

Nonlinear ultrasound imaging leverages harmonic wave generation to enhance contrast and spatial resolution beyond the capabilities of conventional linear techniques. This behavior is commonly modeled by the Westervelt equation, which captures finite-amplitude acoustic wave propagation in heterogeneous media. In this work, we investigate an inverse problem for a periodic nonlinear Westervelt equation in $\mathbb{R}^d$, where $d\in\{2,3\}$ with spatially varying coefficients and Robin-type boundary conditions. The objective is to simultaneously reconstruct the sound speed, diffusivity, and nonlinearity parameters from (partial) boundary measurements. We first establish the Fréchet differentiability of the forward solution operator with respect to the unknown parameters, providing a rigorous analytical foundation for parameter identification. To address uniqueness, we introduce a reference-state framework and prove linearized uniqueness of an all-at-once forward operator without requiring the reference states to satisfy the governing equation. Building on these results, we develop an iterative reconstruction scheme based on a frozen Newton-type method, supported by an exact range invariance property. Numerical simulations are presented to illustrate the feasibility and performance of the proposed approach.

In this paper, we introduce a symmetrization technique for the distributional gradient of a function of bounded variation in the anisotropic setting. This generalizes the result obtained in the Euclidean case in [Amato-Gentile-Nitsch-Trombetti, 2024] by separating the absolutely continuous part of the anisotropic gradient from its singular part. Our main result is an $L^1$ comparison between the function and its anisotropic symmetrization. Moreover, as an application, we derive isoperimetric inequalities for some geometric functionals related to the torsional rigidity.

We prove a large-data $L^2$-decay estimate for nonlinear dissipative Schrödinger equations with attractive-dissipative power nonlinearity. The main difficulty is the lack of sign definiteness of the standard energy when $\Re\lambda<0$, which prevents the usual energy argument from directly yielding a uniform gradient bound. We introduce an augmented energy, obtained by adding a suitable multiple of the decreasing $L^2$-norm to the standard energy. This produces an additional dissipative term and gives a direct uniform-in-time $H^1$ bound without the iteration argument used in previous works. Consequently, for arbitrary initial data in the weighted energy space $\Sigma = H^1 \cap \mathcal{F}H^1$, we obtain the decay rate previously known under the strong dissipative condition throughout the sharp decay range $1<p\le 1+2/d$. This removes the remaining restriction $p\le 1+4/(3d)$ in the attractive-dissipative case.

A classical result due to Frank and Seiringer asserts that for $1\leq p<\frac Ns$, there exists a sharp constant $\mathcal{C}_{N,s,p}>0$ such that $$ \delta_{s,p}(u):=\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}\,dx\,dy-\mathcal{C}_{N,s,p}\int_{\mathbb{R}^N}\frac{|u(x)|^p}{|x|^{sp}}\,dx\ge0, $$ for all $u\in W^{s,p}(\mathbb{R}^N)$. The optimal constant is explicitly known. We investigate quantitative refinements of this inequality. Our first result shows that, under the normalization $ \int_{\mathbb{R}^N}\frac{|u(x)|^p}{|x|^{sp}}\,dx=1,$ the inequality \[ \delta_{s,p}(u)\gtrsim\bigl(\mathrm{dist}_{s,p}(u,\mathcal{Z})\bigr)^\alpha, \] holds, where $\alpha=\max\{4,2p\}$, $\mathcal{Z}$ denotes the family of “virtual” extremals, and the distance is measured in Marcinkiewicz (weak-$L^{p_s^*}$) space. The stability exponent remains constant for $p\le2$, while it depends on $p$ for $p>2$. Our approach is based on a localized Poincaré-Sobolev inequality combined with suitable rescaling and Lorentz embeddings. We exploit a decomposition of the nonlocal energy together with Lorentz estimates, which enables us to control the deficit $\delta_{s,p}(u)$ in terms of the distance to $\mathcal{Z}$. The method also applies to the local case $s=1$, the argument is rearrangement-free and the exponent in the stability estimate improves the existing literature. For $p=2$, via an Emden-Fowler correspondence and pseudo-differential operators, we show that the nonlocal Hardy deficit coincides with the local one and obtain quantitative stability on $\mathbb{R}\times\mathbb{S}^{N-1}$ using the diagonalization of the fractional Hardy quadratic form due to Frank, Lieb, and Seiringer. As an application, we establish a Hardy-Heisenberg-type uncertainty principle in the nonlocal setting, which appears to be new in the literature.

We investigate the stretching mechanism of Finitely Extensible Nonlinear Elastic (FENE) model of polymers in a random turbulent flow. The turbulent model includes a dominant space-scale $\ell\sim N^{-1}$, a dominant time-scale $\tau$, and is white in time. Under suitable scaling assumption, the polymer density equation, initially a stochastic Fokker-Planck equation in the presence of transport-stretching noise, converges weakly as $N\uparrow \infty$ to a limit deterministic equation with a new extra term, a second order operator. This operator, whose shape has been predicted in the physical literature by other arguments, express a sort of average `turbulent stretching' effect. With respect to other derivation of this effective model, the main novelty of our approach is that the deterministic limit is obtained pathwise, without having to take averages with respect to different realizations of the random flow. Next, we consider the limit as $\tau \downarrow 0$ and we identify the stationary distribution of the polymer length. The analysis is carried out in appropriate weighted spaces, which take into account the singularity of the FENE force near the boundary and the no-flux boundary condition, and combines stochastic scaling limit and singular limit techniques.

We prove the existence of global solutions to the nonlinear wave equation in $\mathbb{R}^{1+3}$ $$\Phi_{tt} - \Delta \Phi \pm \Phi|\Phi|^{p-1} = 0$$ in the energy-supercritical regime $p>5$, for a class of large initial data. Our initial data can be decomposed into two pieces, one which is dispersed in the sense of having large $L^2$ norm, while the other piece takes a localised short-pulse form. Consequently, we can obtain global existence for a class of initial data which is large in every homogeneous Sobolev norm $\dot{H}^s_x$ with $s \geq 0$.

We study a one-dimensional thin film equation combining competitive effects of aggregation and repulsion, where repulsion is modeled by fourth-order diffusion and aggregation by backward second-order degenerate diffusion with exponent $m>0$. Under natural regularity constraints, we prove that for every $m>0$, there exists a unique (up to the mass-critical case $m=3$) nonnegative, radially decreasing steady state $U_*$ which coincides with the extremal function of the sharp Sz.-Nagy inequality and is simultaneously the global minimizer of the free energy. Using this variational characterization in the supercritical regime $3<m<\infty$, we show that finite-time blow-up occurs for all initial data whose initial free energy lies below the positive threshold $F(U_*)$, provided the $L^{m+1}$-norm of the initial datum exceeds that of $U_*$. Conversely, if the $L^{m+1}$-norm is below that of $U_*$, the solution exists globally and its second moment diverges as $t\to\infty$. This sharp criterion significantly extends the previously known blow-up condition requiring negative free energy to a much wider class of initial data (see \cite{BP00}). Our results identify the steady state $U_*$ as the critical pivot linking variational structure to dynamical behavior, and provide a constructive method to determine blow-up versus global existence via an explicit $L^{m+1}$-norm comparison.

Travel-time tomography seeks to recover a hidden environment from external measurements generated by propagation through an anomalous region. Standard formulations treat propagation as passive, so the environment influences observations mainly by bending paths or changing travel times. Many collective systems do not operate in that regime: observed arrivals are shaped by strategic motion, congestion, and environmental costs. We formulate this active setting through mean field games, in which the unknown environment enters the running cost through a spatial cost field and observations are read from the resulting population dynamics. This yields three contributions. First, it places propagation, observation generation, and inversion within one PDE-constrained model. Second, it clarifies why the inverse problem differs structurally from passive tomography: kinetic, congestion, and environmental effects are coupled endogenously and appear through space- and time-dependent local weights rather than externally chosen global coefficients. Third, it introduces a two-stage inversion pipeline that combines diffusion-based initialization with full mean field game refinement, and numerical experiments show stable recovery under noise and across multichannel scene families. Taken together, these ingredients establish a foundational framework for a class of inverse problems in which data are generated by optimizing and interacting populations rather than by passive signals. The framework identifies the forward MFG model, admissible observation channels, structural mechanisms, and computational recovery route needed to study active tomography under collective dynamics. Such problems arise, among other settings, in biological transport, vascular flow, and subsurface groundwater dynamics.

We investigate the obstacle problem for generalized Dean–Kawasaki equations driven by correlated conservative noise, establishing the existence, uniqueness, and $L^1$-stability of stochastic kinetic solutions. Our core strategy combines a kinetic characterization of the Skorokhod condition with a precise description of the reflection measure term associated with the obstacle, in which the barrier substitutes the solution. This formulation makes the reflection mechanism explicit at the kinetic level and yields a stable framework adapted to $L^1$ doubling of variables method. Consequently, under a merely continuous obstacle and the same structural assumptions as in the obstacle-free setting, we obtain well-posedness over the full porous-medium regime, covering degenerate diffusion and the critical square-root noise coefficient. This extends the existing theory of obstacle problems for stochastic partial differential equations to a class of degenerate equations with singular diffusion coefficients.

In this paper, we are concerned with local controllability properties of degenerate parabolic equations in bounded domains that evolve in time. More precisely, we deal with the exact controllability to a positive trajectory of a one-dimensional semilinear degenerate equation governed via the coefficient of the reaction term. We apply a well-known local inversion method combined with some appropriate specific estimates.

In this paper we construct a self-similar fractal configured as an infinitely branched tree and equip it with a regular self-similar Dirichlet form. We show anomalous behaviour of the mean exit time with respect to typical metric balls. Under properly selected self-similar measure, we further show the weak elliptic Harnack inequality holds but the strong analogue fails.

In this paper, we establish symmetry results for solutions of the mean field equation \[ \frac{\alpha}{2} \Delta u + e^u - 1 = 0 \] on $ \mathbb{S}^2 $ for $\frac{1}{3}\leq \alpha \leq 1$, under a geometric condition $(\mathcal{H})$ introduced below. The proofs utilize the Sphere Covering Inequality and incorporate topological arguments on $ \mathbb{S}^2 $. These results are further applied to demonstrate a rigidity property of the Hawking mass for stable constant mean curvature (CMC) spheres, addressing a question posed by Robert Bartnik in 2002. Our results unify and extend previous rigidity results obtained under symmetry or hemispherical balance assumptions, and apply beyond the nearly spherical setting.

We use convex integration techniques to provide examples of failure of weighted Calderón-Zygmund estimates for degenerate linear elliptic PDEs when the weights are in $A_p$, $p > 2$.

Low-rank methods for kinetic equations have attracted increasing attention due to their effectiveness in reducing the high dimensionality of phase space. In our previous work [G. Wang & J. Hu, J. Comput. Phys. 558 (2026) 114884], we developed a dynamical low-rank method based on the projector-splitting integrator in tensor-train (TT) format, in which explicit time integration is employed in all substeps. As a result, the method is subject to severe stability constraints in the strongly collisional regimes. In this paper, we consider kinetic equations with the (nonlinear) Fokker–Planck collision operator and develop a dynamical low-rank method that employs implicit or implicit-explicit (IMEX) discretizations in appropriate substeps to overcome stiffness. In these implicit substeps, the resulting equations can be formulated as matrix or tensor Sylvester equations, for which we propose efficient direct solvers by exploiting their underlying structure. The overall computational cost of the proposed method scales linearly with respect to the number of grid points in a single velocity dimension, comparable to that of a fully explicit low-rank scheme. We demonstrate the accuracy and efficiency of the proposed method on several representative kinetic test problems.

We extend the De Giorgi–Nash–Moser theory to superposition operators of mixed fractional operators. In particular, we investigate several regularity properties for this class of operators. We establish the Caccioppoli-type inequality with tail for weak subsolutions, local boundedness of weak subsolutions, local Hölder continuity of weak solutions, the weak Harnack inequality for weak supersolutions, and the lower semicontinuity of weak supersolutions. Furthermore, we prove the expansion of positivity, a preliminary Harnack inequality, and the upper semicontinuity of weak subsolutions. Our results apply to both fixed-sign and sign-changing solutions involving mixed local–nonlocal superposition fractional operators. Notably, the results are new even in the classical linear case $p=2$, demonstrating the broader applicability of the techniques developed in this work.

We consider a one-parameter family of 1D models for the 3D axisymmetric incompressible Euler equation with $C^{\alpha}$ vorticity and without swirl near the symmetry axis. For $\alpha = \frac13$, we impose a crucial normalization and construct a $C^{\infty}$ self-similar blowup profile with unbounded 1D stream function and infinite spatial blowup rate, using a fixed-point argument around a numerically constructed approximate profile. For $\alpha < \frac13$ sufficiently close to $\frac13$, we perturb the $\frac13$-profile and analytically construct exact smooth 1D profiles with bounded stream function and finite spatial blowup rate. In the companion work~\cite{chen2026eulerII}, for any $\alpha \in (0,\frac13)$, we lift these 1D blowup profiles to construct exact $C^{1,\alpha}$ self-similar blowup profiles for 3D Euler, and build on them to prove sharp asymptotically self-similar blowup for 3D axisymmetric Euler without swirl from $C_c^\alpha$ initial vorticity and $C^{1,\alpha} \cap L^2$ initial velocity.

Conservation laws of a class of time-dependent damped nonlinear multidimensional wave equations are derived by Noether's theorem. For arbitrary nonzero damping coefficient and nonlinear interaction term, its infinitesimal variational symmetries span a Euclidean algebra $\euclid(n)$ of space translations and rotations. They produce conservation of linear and angular momentums. For some specific forms of these two terms symmetry algebra is enlarged to a subalgebra of the conformal algebra $\conf(1,n)$ and in this case more interesting conservation laws are found.