L p Asymptotics for the Heat Equation on Symmetric Spaces for Non-symmetric Solutions - Papageorgiou, Efthymia (International Mathematics Research Notices)
Abstract The main goal of this work is to study the $L^{p}$-asymptotic behavior of solutions to the heat equation on arbitrary rank Riemannian symmetric spaces of non-compact-type $G/K$ for non-bi-$K$ invariant initial data. For initial data $u_{0}$ compactly supported or in a weighted $L^{1}(G/K)$ space with a weight depending on $p\in [1, \infty ]$, we introduce a mass function $M_{p}(u_{0})(\cdot )$, and prove that if $h_{t}$ is the heat kernel on $G/K$, then $$ \begin{align*} &\|h_t\|_p^{-1}\,\|u_0\ast h_t \, - \,M_p(u_0)(\cdot)\,h_t\|_p \rightarrow 0 \quad \textrm{as} \quad t\rightarrow \infty.\end{align*} $$ Interestingly, the $L^{p}$ heat concentration leads to completely different expressions of the mass function for $1\leq p <2$ and $2\leq p\leq \infty $. If we further assume that the initial data are bi-$K$-invariant, then our mass function boils down to the constant $\int _{G/K}u_{0}$ in the case $p=1$, and more generally to $\mathcal{H}{u_{0}}(i\rho (2/p-1))$ if $1\leq p<2$, and to $\mathcal{H}{u_{0}}(0)$ if $2\leq p \leq \infty $. Thus, we improve upon results by Vázquez, Anker et al., and Naik et al., clarifying the nature of the problem.

A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System - Winkler, Michael (International Mathematics Research Notices)
Abstract The Neumann problem for (0.1)$$ \begin{align}& V_t = \Delta V-aV+f(x,t) \end{align}$$is considered in bounded domains $\Omega \subset {\mathbb {R}}^n$ with smooth boundary, where $n\ge 1$ and $a\in {\mathbb {R}}$. By means of a variational approach, a statement on boundedness of the quantities $$ \begin{eqnarray*} \sup_{t\in (0,T)} \int_\Omega \big|\nabla V(\cdot,t)\big|^p L^{\frac{n+p}{n+2}} \Big( \big|\nabla V(\cdot,t)\big| \Big) \end{eqnarray*}$$in dependence on the expressions (0.2)$$ \begin{align}& \sup_{t\in (0,T-\tau)} \int_t^{t+\tau} \int_\Omega |f|^{\frac{(n+2)p}{n+p}} L\big( |f|\big) \end{align}$$is derived for $p\ge 2$, $\tau>0$, and $T\ge 2\tau $, provided that $L\in C^0([0,\infty ))$ is positive, strictly increasing, unbounded, and slowly growing in the sense that $\limsup _{s\to \infty } \frac {L(s^{\lambda _0})}{L(s)} <\infty $ for some $\lambda _0>1$. In the particular case when $p=n\ge 2$, an additional condition on growth of $L$, particularly satisfied by $L(\xi ):=\ln ^\alpha (\xi +b)$ whenever $b>0$ and $\alpha>\frac {(n+2)(n-1)}{2n}$, is identified as sufficient to ensure that as a consequence of the above, bounds for theintegrals in (0.2) even imply estimates for the spatio-temporal modulus of continuity of solutions to (0.1). A subsequent application to the Keller–Segel system $$ \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot \big( D(v)\nabla u\big) - \nabla \cdot \big( uS(v)\nabla v\big) + ru - \mu u^2, \\[1mm] v_t = \Delta v-v+u, \end{array} \right. \end{eqnarray*}$$shows that when $n=2$, $r\in {\mathbb {R}}$, $0<D\in C^2([0,\infty ))$, and $S\in C^2([0,\infty )) \cap W^{1,\infty }((0,\infty ))$ and thus especially in the presence of arbitrarily strong diffusion degeneracies implied by rapid decay of $D$, any choice of $\mu>0$ excludes blowup in the sense that for all suitably regular nonnegative initial data, an associated initial-boundary value problem admits a global bounded classical solution.
Keywords: General Mathematics

A Result on Parabolic Gradient Regularity in Orlicz Spaces and Application to Absorption-Induced Blow-Up Prevention in a Keller–Segel-Type Cross-Diffusion System - Winkler, Michael (International Mathematics Research Notices)
Abstract The Neumann problem for (0.1)$$ \begin{align}& V_t = \Delta V-aV+f(x,t) \end{align}$$is considered in bounded domains $\Omega \subset {\mathbb {R}}^n$ with smooth boundary, where $n\ge 1$ and $a\in {\mathbb {R}}$. By means of a variational approach, a statement on boundedness of the quantities $$ \begin{eqnarray*} \sup_{t\in (0,T)} \int_\Omega \big|\nabla V(\cdot,t)\big|^p L^{\frac{n+p}{n+2}} \Big( \big|\nabla V(\cdot,t)\big| \Big) \end{eqnarray*}$$in dependence on the expressions (0.2)$$ \begin{align}& \sup_{t\in (0,T-\tau)} \int_t^{t+\tau} \int_\Omega |f|^{\frac{(n+2)p}{n+p}} L\big( |f|\big) \end{align}$$is derived for $p\ge 2$, $\tau>0$, and $T\ge 2\tau $, provided that $L\in C^0([0,\infty ))$ is positive, strictly increasing, unbounded, and slowly growing in the sense that $\limsup _{s\to \infty } \frac {L(s^{\lambda _0})}{L(s)} <\infty $ for some $\lambda _0>1$. In the particular case when $p=n\ge 2$, an additional condition on growth of $L$, particularly satisfied by $L(\xi ):=\ln ^\alpha (\xi +b)$ whenever $b>0$ and $\alpha>\frac {(n+2)(n-1)}{2n}$, is identified as sufficient to ensure that as a consequence of the above, bounds for theintegrals in (0.2) even imply estimates for the spatio-temporal modulus of continuity of solutions to (0.1). A subsequent application to the Keller–Segel system $$ \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot \big( D(v)\nabla u\big) - \nabla \cdot \big( uS(v)\nabla v\big) + ru - \mu u^2, \\[1mm] v_t = \Delta v-v+u, \end{array} \right. \end{eqnarray*}$$shows that when $n=2$, $r\in {\mathbb {R}}$, $0<D\in C^2([0,\infty ))$, and $S\in C^2([0,\infty )) \cap W^{1,\infty }((0,\infty ))$ and thus especially in the presence of arbitrarily strong diffusion degeneracies implied by rapid decay of $D$, any choice of $\mu>0$ excludes blowup in the sense that for all suitably regular nonnegative initial data, an associated initial-boundary value problem admits a global bounded classical solution.

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces - Küster, Benjamin; Weich, Tobias (International Mathematics Research Notices)
Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{τ }$ over the unit cosphere bundle $S^{* }\mathcal M$, we give a precise description of those classical (Pollicott–Ruelle) resonant states on $\mathbf V_{τ }$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^{* }\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,σ )$ on compatible associated vector bundles $\mathbf W_{σ }$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott–Ruelle spectrum. Further, under some mild assumptions on the representations $τ$ and $σ$ defining the bundles $\mathbf V_{τ }$ and $\mathbf W_{σ }$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott–Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_{σ }$. Our methods of proof are based on representation theory and Lie theory.
Keywords: General Mathematics

Sonine Formulas and Intertwining Operators in Dunkl Theory - Rösler, Margit; Voit, Michael (International Mathematics Research Notices)
Abstract Let $V_k$ denote Dunkl’s intertwining operator associated with some root system $R$ and multiplicity $k$. For two multiplicities $k, k^{′ }$ on $R$, we study the intertwiner $V_{k^{′ },k} = V_{k^{′ }}° V_k^{-1}$ between Dunkl operators with multiplicities $k$ and $k^{′ }.$ It has been a long-standing conjecture that $V_{k^{′ },k}$ is positive if $k^{′ } ≥ k ≥ 0.$ We disprove this conjecture by constructing counterexamples for root system $B_n$. This matter is closely related to the existence of Sonine-type integral representations between Dunkl kernels and Bessel functions with different multiplicities. In our examples, such Sonine formulas do not exist. As a consequence, we obtain necessary conditions on Sonine formulas for Heckman–Opdam hypergeometric functions of type $BC_n$ and conditions for positive branching coefficients between multivariable Jacobi polynomials.
Keywords: General Mathematics

Locally complete intersection maps and the proxy small property - Briggs, Benjamin; Iyengar, Srikanth B.; Letz, Janina Carmen; Pollitz, Josh (Int. Math. Res. Not. IMRN)

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces - Küster, Benjamin; Weich, Tobias (International Mathematics Research Notices)
Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{τ }$ over the unit cosphere bundle $S^{* }\mathcal M$, we give a precise description of those classical (Pollicott–Ruelle) resonant states on $\mathbf V_{τ }$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^{* }\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,σ )$ on compatible associated vector bundles $\mathbf W_{σ }$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott–Ruelle spectrum. Further, under some mild assumptions on the representations $τ$ and $σ$ defining the bundles $\mathbf V_{τ }$ and $\mathbf W_{σ }$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott–Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_{σ }$. Our methods of proof are based on representation theory and Lie theory.
Keywords: General Mathematics

Can Rotational Fluxes Impede the Tendency Toward Spatial Homogeneity in Nutrient Taxis(-Stokes) Systems? - Winkler, Michael (International Mathematics Research Notices)
Abstract We consider the spatially 2D version of the model $$\begin{equation*} \qquad\quad\left\{ \begin{array}{@{}rcll} n_t + u\cdot\nabla n &=& \Delta n - \nabla \cdot \big(nS(x,n,c) \cdot \nabla c \big), \qquad &\qquad x\in \Omega, \ t>0, \\ c_t + u\cdot \nabla c &=& \Delta c - n f(c), \qquad &\qquad x\in \Omega, \ t>0, \\ u_t &=& \Delta u + \nabla P + n\nabla\phi, \qquad \nabla\cdot u=0, \qquad &\qquad x\in \Omega, \ t>0, \end{array} \right. \qquad \qquad (\star) \end{equation*}$$for nutrient taxis processes, possibly interacting with liquid environments. Here the particular focus is on the situation when the chemotactic sensitivity $S$ is not a scalar function but rather attains general values in ${\mathbb{R}}^{2\times 2}$, thus accounting for rotational flux components in accordance with experimental findings and recent modeling approaches. Reflecting significant new challenges that mainly stem from apparent loss of energy-like structures, especially for initial data with large size, the knowledge on ($\star$) so far seems essentially restricted to results on global existence of certain generalized solutions with possibly quite poor boundedness and regularity properties; widely unaddressed seem aspects related to possible effects of such non-diagonal taxis mechanisms on the qualitative solution behavior, especially with regard to the fundamental question whether spatial structures may thereby be supported. The present work answers the latter in the negative in the following sense: under the assumptions that the initial data $(n_0,c_0,u_0)$ and the parameter functions $S$, $f$, and $\phi$ are sufficiently smooth, and that $S$ is bounded and $f$ is positive on $(0,\infty )$ with $f(0)=0$, it is shown that any nontrivial of these solutions eventually becomes smooth and satisfies $$\begin{equation*} n(\cdot,t)\to - \int_\Omega n_0, \quad c(\cdot,t)\to 0 \quad \text{and} \quad u(\cdot,t)\to 0 \qquad \text{as} \ t\to\infty, \end{equation*}$$uniformly with respect to $x\in \Omega$. By not requiring any smallness condition on the initial data, the latter seems new even in the corresponding fluid-free version obtained on letting $u\equiv 0$ in ($\star$).

Faithful Realizability of Tropical Curves - Cheung, Man-Wai; Fantini, Lorenzo; Park, Jennifer; Ulirsch, Martin (International Mathematics Research Notices)

On p-adic L-functions for GL(n) × GL (n-1) over totally real fields - Januszewski, Fabian (International Mathematics Research Notices)
Keywords: General Mathematics

The Heat Semigroup in the Compact Heckman-Opdam Setting and the Segal-Bargmann Transform - Rösler, Margit; Remling, H. (International Mathematics Research Notices)
Keywords: General Mathematics

Weighted Distribution of the 4-rank of Class Groups and Applications - Fouvry, Étienne; Klüners, Jürgen (International Mathematics Research Notices)
We prove that the distribution of the values of the 4-rank of ideal class groups of quadratic fields is not affected when it is weighted by a divisor type function. We then give several applications concerning a new lower bound of the sums of class numbers of real quadratic fields with discriminant less than a bound tending to infinity and several questions of P. Sarnak concerning reciprocal geodesics.
Keywords: General Mathematics

Positivity of Dunkl's intertwining operator via the trigonometric setting - Rösler, Margit; Voit, Michael (International Mathematics Research Notices)
In this note, a new proof for the positivity of Dunkl's intertwining operator in the crystallographic case is given. It is based on an asymptotic relationship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, and on positivity results of S. Sahi for the Heckman-Opdam polynomials and their non-symmetric counterparts.