The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large-time asymptotics

In this paper, we consider the classical Riemann problem for a generalized Burgers equation, $$u_t+h_{\alpha }(x)u{u}_x=u_{xx},$$ with a spatially dependent, nonlinear sound speed, $h_{\alpha }(x)\equiv {(1+x^2)}^{-\alpha }$ with $\alpha >0$, which decays algebraically with increasing distance from a fixed spatial origin. When $\alpha =0$, this reduces to the classical Burgers equation. In this first part of a pair of papers, we focus attention on the large-time structure of the associated Riemann problem, and obtain its detailed structure, as $t\to \infty$, via the method of matched asymptotic coordinate expansions (this uses the classical method of matched asymptotic expansions, with the asymptotic parameters being the independent coordinates in the evolution problem; this approach is developed in detail in the monograph of Leach and Needham, as referenced in the text), over all parameter ranges. We identify a significant bifurcation in structure at $\alpha =\frac{1}{2}$.     

Properties of given and detected unbounded solutions to a class of chemotaxis models

This paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction–repulsion model, with nonlinear productions, diffusion, sensitivities, and logistic term, we detect Lebesgue spaces where given unbounded solutions also blow up in the corresponding norms of those spaces; subsequently, estimates for the blow-up time are established. Finally, for a simplified version of the model, some blow-up criteria are proved. More precisely, we analyze a zero-flux chemotaxis system essentially described as (⋄) The problem is formulated in a bounded and smooth domain Ω of ${\mathbb{R}}^n$, with $n\ge 1$, for some $m_1,m_2,m_3\in \mathbb{R}$, $\chi ,\xi ,\alpha ,\beta ,\lambda ,\mu >0$, $k>1$, and with $T_{max}\in (0,\infty ]$. A sufficiently regular initial data $u_0\ge 0$ is also fixed. Under specific relations involving the above parameters, one of these always requiring some largeness conditions on $m_2+\alpha$,

• (i) we prove that any given solution to ($\diamond$), blowing up at some finite time $T_{max}$ becomes also unbounded in $L^{\mathfrak{p}}(\mathrm{\Omega })$-norm, for all $\mathfrak{p}>\frac{n}{2}(m_2-m_1+\alpha )$;
• (ii) we give lower bounds T (depending on ${\int }_{\mathrm{\Omega }}{u}_0^{\overline{p}}$) of $T_{max}$ for the aforementioned solutions in some $L^{\overline{p}}(\mathrm{\Omega })$-norm, being $\overline{p}=\overline{p}(n,m_1,m_2,m_3,\alpha ,\beta )\ge \mathfrak{p}$;
• (iii) whenever $m_2=m_3$, we establish sufficient conditions on the parameters ensuring that for some u₀ solutions to ($\diamond$) effectively are unbounded at some finite time.

Within the context of blow-up phenomena connected to problem ($\diamond$), this research partially improves the analysis in Wang et al. (J Math Anal Appl. 2023;518(1):126679) and, moreover, contributes to enrich the level of knowledge on the topic.     

Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals

We consider the highly oscillatory integral $F(w):={\int }_{-\infty }^{\infty }{e}^{iw(t^{K+2}+e^{i\theta }{t}^p)}g(t)dt$ for large positive values of w, , K and p positive integers with $1\le p\le K$, and $g(t)$ an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when $w\to +\infty$ for general values of K and p in terms of elementary functions, and determine the Stokes lines. For $p\ne 1$, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters K and p; the special case $p=1$ requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals ${\mathrm{\Psi }}_K(x_1,x_2,\text{…},x_K)$ for large values of one of its variables, say $x_p$, and bounded values of the remaining ones. This family of integrals may be written in the form $F(w)$ for appropriate values of the parameters w, θ and the function $g(t)$. Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large $|x_p|$. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.     

Multiple orthogonal polynomials associated with the exponential integral

We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights $(w_1,w_2)$ on the positive real line, with $w_1(x)=x^{\alpha }{e}^{-x}$ the gamma density and $w_2(x)=x^{\alpha }{E}_{\nu +1}(x)$ a density related to the exponential integral $E_{\nu +1}$. We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed-type multiple orthogonal polynomials.     

Criterion of Bari basis property for 2 × 2 Dirac-type operators with strictly regular boundary conditions

The paper is concerned with the Bari basis property of a boundary value problem associated in $L^2([0,1];{\mathbb{C}}^2)$ with the following 2 × 2 Dirac-type equation for $y=col(y_1,y_2)$: with a potential matrix $Q\in L^2([0,1];{\mathbb{C}}^{2\times 2})$ and subject to the strictly regular boundary conditions $Uy:=\{U_1,U_2\}y=0$. If $b_2=-b_1=1$, this equation is equivalent to one-dimensional Dirac equation. We show that the normalized system of root vectors ${\{f_n\}}_{n\in \mathbb{Z}}$ of the operator $L_U(Q)$ is a Bari basis in $L^2([0,1];{\mathbb{C}}^2)$ if and only if the unperturbed operator $L_U(0)$ is self-adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result: Let $Q\in L^p([0,1];{\mathbb{C}}^{2\times 2})$, $p\in [1,2]$, boundary conditions be strictly regular, and let ${\{g_n\}}_{n\in \mathbb{Z}}$ be the sequence biorthogonal to the normalized system of root vectors ${\{f_n\}}_{n\in \mathbb{Z}}$ of the operator $L_U(Q)$. Then, $$\{\parallel f_n-g_n{{\parallel }_2\}}_{n\in \mathbb{Z}}\in {({\ell }^p(\mathbb{Z}))}^{\ast }\iff L_U(0)=L_U{(0)}^{\ast }.$$ These abstract results are applied to noncanonical initial-boundary value problem for a damped string equation.     

On the n-th linear polarization constant of Rn$\mathbb {R}^n$

We prove that given any set of n unit vectors ${\{v_i\}}_{i=1}^n\subset {\mathbb{R}}^n$, the inequality $$\underset{{\parallel x\parallel }_{{\mathbb{R}}^n}=1}{sup}|⟨x,v_1⟩\cdots ⟨x,v_n⟩|\ge n^{-n/2}$$ holds for $n\le 14$. Moreover, the equality is attained if and only if ${\{v_i\}}_{i=1}^n$ is an orthonormal system.     

Quantifying properties (K) and (μs$\mu ^{s}$)

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence ${(f_n)}_{n=1}^{\infty }$ so that $⟨f_n,x_n⟩\to 0$ as $n\to \infty$ for every weakly null sequence ${(x_n)}_{n=1}^{\infty }$ in X; X has property $({\mu }^s)$ if every weak* null sequence in $X^{\ast }$ admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property $({\mu }^s)$ and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.     

Linearizable Abel equations and the Gurevich–Pitaevskii problem

Applying symmetry reduction to a class of $\mathrm{SL}(2,\mathbb{R})$-invariant third-order ordinary differential equations (ODEs), we obtain Abel equations whose general solution can be parameterized by hypergeometric functions. Particular case of this construction provides a general parametric solution to the Kudashev equation, an ODE arising in the Gurevich–Pitaevskii problem, thus giving the first term of a large-time asymptotic expansion of its solution in the oscillatory (Whitham) zone.     

Boundary behavior of the solution to the linear Korteweg-De Vries equation on the half line

In this paper, we consider the solution to the linear Korteweg-De Vries (KdV) equation, both homogeneous and forced, on the quadrant $\{x\in {\mathbb{R}}^{+},t\in {\mathbb{R}}^{+}\}$ via the unified transform method of Fokas and we provide a complete rigorous study of the integrals of the formula provided by the method, especially focusing on the explicit verification of the considered initial-boundary-value problems (IBVPs), with generic data, as well as on the uniform convergence of all its derivatives, as $(x,t)$ approaches the boundary of the quadrant, and their rapid decay as $x\to \infty$.     

Stability of stationary solutions to the Navier–Stokes equations in the Besov space

We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space ${\mathbb{R}}^n$ for $n\ge 3$. It is clarified that if w is small in ${\dot{B}}_{p_{*},q^{\prime }}^{-1+\frac{n}{p_{*}}}$ for and , then for every small initial disturbance $a\in {\dot{B}}_{p_0,q}^{-1+\frac{n}{p_0}}$ with and ($1/q+1/q^{\prime }=1$), there exists a unique solution $v(t)$ of the nonstationary Navier–Stokes equations on (0, ∞) with $v(0)=w+a$ such that ${\parallel v(t)-w\parallel }_{L^r}=O(t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{r})})$ and ${\parallel v(t)-w\parallel }_{{\dot{B}}_{p,q}^s}=O(t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{p})-\frac{s}{2}})$ as $t\to \infty$, for , , and small $s>0$.     

On the interpolation constants for variable Lebesgue spaces

For $\theta \in (0,1)$ and variable exponents $p_0(·),q_0(·)$ and $p_1(·),q_1(·)$ with values in [1, ∞], let the variable exponents $p_{\theta }(·),q_{\theta }(·)$ be defined by $$1/p_{\theta }(·):=(1-\theta )/p_0(·)+\theta /p_1(·),1/q_{\theta }(·):=(1-\theta )/q_0(·)+\theta /q_1(·).$$ The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space $L^{p_j(·)}$ to the variable Lebesgue space $L^{q_j(·)}$ for $j=0,1$, then $${\parallel T\parallel }_{L^{p_{\theta }(·)}\to L^{q_{\theta }(·)}}\le {C\parallel T\parallel }_{L^{p_0(·)}\to L^{q_0(·)}}^{1-\theta }{\parallel T\parallel }_{L^{p_1(·)}\to L^{q_1(·)}}^{\theta },$$ where C is an interpolation constant independent of T. We consider two different modulars ${\varrho }^{\max }(·)$ and ${\varrho }^{\mathrm{sum}}(·)$ generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants C_max and C_sum, which imply that $C_{\max }\le 2$ and $C_{\mathrm{sum}}\le 4$, as well as, lead to sufficient conditions for $C_{\max }=1$ and $C_{\mathrm{sum}}=1$. We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that $p_j(·)=q_j(·)$, $j=0,1$ are Lipschitz continuous and bounded away from one and infinity (in this case, ${\varrho }^{\max }(·)={\varrho }^{\mathrm{sum}}(·)$).     

Three-dimensional lattice ground states for Riesz and Lennard-Jones–type energies

The Riesz potential $f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard-Jones–type potentials $f_{n,m}^{\mathrm{LJ}}(r):=a{r}^{-n}-b{r}^{-m}$, $n>m$ that are widely used in molecular simulations. In this paper, we investigate analytically and numerically the minimizers among three-dimensional lattices of Riesz and Lennard-Jones energies. We discuss the minimality of the body-centered-cubic (BCC) lattice, face-centered-cubic (FCC) lattice, simple hexagonal (SH) lattices, and hexagonal close-packing (HCP) structure, globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for and the HCP lattice is computed to have higher energy than the FCC (for $s>3/2$) and BCC (for ) lattices. In the Lennard-Jones case with exponents , the ground state among lattices is confirmed to be an FCC lattice whereas an HCP phase occurs once added to the investigated structures. Furthermore, phase transitions of type “FCC-SH” and “FCC-HCP-SH” (when the HCP lattice is added) as the inverse density V increases are observed for a large spectrum of exponents $(n,m)$. In the SH phase, the variation of the ratio Δ between the interlayer distance d and the lattice parameter a is studied as V increases. In the critical region of exponents , the SH phase with an extreme value of the anisotropy parameter Δ dominates. If one limits oneself to rigid lattices, the BCC-FCC-HCP phase diagram is found. For , the BCC lattice is the only energy minimizer. Choosing , the FCC and SH latices become minimizers.     

Weyl families of transformed boundary pairs

Let $(\mathfrak{L},\mathrm{\Gamma })$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space $\mathfrak{H}$. Let $M_{\mathrm{\Gamma }}$ be the Weyl family corresponding to $(\mathfrak{L},\mathrm{\Gamma })$. We cope with two main topics. First, since $M_{\mathrm{\Gamma }}$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation $M_{\mathrm{\Gamma }}(z)$, for some $z\in \mathbb{C}\setminus \mathbb{R}$, becomes a nontrivial task. Regarding $M_{\mathrm{\Gamma }}(z)$ as the (Shmul'yan) transform of $zI$ induced by Γ, we give conditions for the equality in $\overline{M_{\mathrm{\Gamma }}(z)}\subseteq \overline{M_{\overline{\mathrm{\Gamma }}}(z)}$ to hold and we compute the adjoint $M_{\overline{\mathrm{\Gamma }}}{(z)}^{\ast }$. As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for $T^{+}$ is nonempty. Based on the criterion for the closeness of $M_{\mathrm{\Gamma }}(z)$, we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family $M_{\mathrm{\Gamma }}$ corresponding to a boundary relation Γ for $T^{+}$ is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$ with its Weyl family $M_{{\mathrm{\Gamma }}^{\prime }}$. The transformation scheme is either ${\mathrm{\Gamma }}^{\prime }=\mathrm{\Gamma }{V}^{-1}$ or ${\mathrm{\Gamma }}^{\prime }=V\mathrm{\Gamma }$ with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between $(\mathfrak{L},\mathrm{\Gamma })$ and $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$; the formula for $M_{{\mathrm{\Gamma }}^{\prime }}-M_{\mathrm{\Gamma }}$, for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple $(\mathfrak{L},{\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1)$ with $ker\mathrm{\Gamma }=T$ and $T_0=T_0^{\ast }$ (second scheme, Hilbert space case).