Papers

We establish the asymptotic stability of planar viscous shock waves for the outflow problem of the three-dimensional barotropic compressible Navier–Stokes equations in a half-space, with periodic boundary conditions imposed in the transverse directions. For a weak shock located sufficiently far from the boundary, we prove that, under small perturbations in $H^{2}$, the outflow problem admits a unique global-in-time solution and that the solution converges uniformly to the viscous shock, up to a dynamical shift, whose velocity time-asymptotically decays. This provides the first shock-stability result for the multidimensional outflow problem. Our proof combines the $a$-contraction method and higher-order energy estimates adapted to the half-space boundary. The boundary trace remaining in the zeroth-order estimate is controlled by differentiating the perturbation system in the tangential directions and exploiting the favorable sign of the outflow flux. The highest-order normal derivatives are then recovered from the time and tangential derivatives through the momentum equation, while the boundary terms generated by the dynamical shift are controlled using the exponential decay of the viscous shock tail.

We study the long-time dynamics of small solutions to the one-dimensional nonlinear Schrödinger equation \[ i\partial_t v+\partial_x^2v-\beta\abs{v}^2v +\cW(x)\abs{v}^4v+\gamma i\partial_xv=0, \] where $\cW$ is spatially localized. The cubic nonlinearity is long range and produces the logarithmic phase correction, whereas the localized quintic term is short range at leading order. We construct a global forward modified wave operator for small complex asymptotic profiles and prove quantitative final-state estimates. For small data in the weighted energy space, we also establish global existence, sharp $t^{-1/2}$ decay, and forward modified scattering with a unique asymptotic profile. The principal new phenomenon occurs beyond this leading law. The exact Duhamel tail generated by the localized quintic term admits a quantitative inner scaling limit at the distinguished frequency $\zeta=-\gamma/2$ on the scale $\abs{\zeta+\gamma/2}\sim t^{-1/2}$. Its universal shape is explicit and depends on the value of the scattering profile on the distinguished ray and on the zeroth moment of $\cW$. When both quantities are nonzero, the limit is nontrivial, belongs optimally to $C^{2,1}_{\mathrm{loc}}$, and is not $C^3$ at the center. Away from the corresponding self-similar ray $\xi=-\gamma$, we construct rigorously defined higher-order outer expansions through every integer order allowed by the decay of $\cW$, and to each fixed finite order when $\cW$ is rapidly decreasing.

We establish the local-in-time existence and uniqueness of monotone solutions to the two-dimensional nonstationary boundary-layer equations for a dilatant power-law fluid in a periodic half-space for $1<n<\frac{7}{3}$. Under Oleinik's monotonicity condition and suitable weighted Sobolev assumptions on the initial data and outer flow, we construct solutions through tangential regularization and derive uniform a priori estimates. A suitable good unknown compensates for the loss of one tangential derivative caused by the normal velocity. By combining weighted energy estimates, the Faà di Bruno formula, and the maximum and minimum principles, we control the nonlinear degenerate diffusion term $\partial_y^2(\omega^n)$, whose effective diffusion coefficient vanishes as $\omega=\partial_yu$ decays at infinity, and propagate the weighted monotonicity of the vorticity. Within this exponent range, our result partially resolves the eleventh open problem posed by Oleinik and Samokhin \cite{OAO} on the existence and uniqueness of solutions to nonstationary boundary-layer systems for dilatant fluids.

We establish gradient potential estimates for solutions obtained as limits of approximations (SOLA) to the fractional $p$-Laplace equation with finite signed Radon measure data. Under the assumptions $n\ge2$, $p>2$, $0<s<1$, and $sp>p-1$, every SOLA belongs to $W^{1,p-1}_{\mathrm{loc}}$, and its weak gradient satisfies a Wolff potential estimate at every Lebesgue point of the weak gradient. Under the additional condition $sp>n$, the solution has a continuous representative that is Fréchet differentiable at every point where the potential is finite. These results give a partial answer to questions raised by Diening and Nowak [Ann. PDE, 11(2025)] and by Diening, Kim, Lee and Nowak~[J. Eur. Math. Soc. (JEMS), 2025] concerning gradient potential estimates for the fractional $p$-Laplacian. The proof combines homogeneous affine decay with constants independent of the affine slope and comparison estimates to obtain an affine excess recurrence. Iteration then yields the Wolff potential bound.

We investigate the dynamical behavior of a class of semilinear stochastic wave equations on a bounded smooth domain $\Ocal\subset\R^d$ driven by additive trace-class noise, where the elastic response is governed by a mixed local–nonlocal operator $\Acal=-\theta\Delta+\beta(-\Delta)^s$ with $s\in(0,1)$. A fundamental challenge in this setting is that the local and nonlocal operators do not commute on bounded domains: the natural Dirichlet basis fails to diagonalize the restricted fractional Laplacian. Consequently, we first establish the strong convergence of the resulting non-diagonal, dense Galerkin approximation scheme. Leveraging these uniform energy bounds, we rigorously derive the associated Itô energy identity. In the defocusing regime ($\varepsilon = +1$), this strong approximation yields global well-posedness on the energy-subcritical range, providing a unique probabilistically strong solution in the energy space $V \times H$. Within this variational framework, we conduct an analysis of the Malliavin regularity, showing $(u(t), v(t)) \in \mathbb{D}^{1,2}(V) \times \mathbb{D}^{1,2}(H)$, and leverage fractional Sobolev embeddings to prove that the one-dimensional probability law of $u(t, x_0)$ is absolutely continuous via the Bouleau–Hirsch criterion. In stark contrast, for the focusing regime ($\varepsilon = -1$), we establish local well-posedness and prove a rigorous dichotomy: under a negativity condition on the initial energy, either pathwise explosion occurs with positive probability in finite time, or the energy norm possesses an infinite second moment before an explicit critical time $T^*$. Finally, we observe how the dense Galerkin interaction matrices pose unique structural challenges for spatial statistical inference.

We investigate weak solutions to the two-dimensional quasi-geostrophic equations without dissipation. We establish global existence of weak solutions for temperature bounded and lacking spatial decay and velocity in the space $L^2_{ul}(\mathbb{R}^2)$. Our methods rely on a spectral Serfati identity, which we use to establish uniform $L^2_{ul}$ bounds on a sequence of velocities satisfying the dissipative equations. These bounds, combined with a maximum principle on the scalar temperature, allow us to pass to the zero-dissipation limit, giving global-in-time weak solutions.

We settle the planar sharp Hessian-integrability conjecture of Armstrong, Silvestre, and Smart [Comm. Pure Appl. Math. 65 (2012), 1169–1184] for viscosity supersolutions of fully nonlinear uniformly elliptic equations. If $\kappa=\Lambda/\lambda$, then the optimal exponent in the $W^{2,\epsilon}$ regularity theory in the plane is exactly $$ \epsilon_2(\kappa)=\frac{2}{\kappa+1}. $$ The same mechanism reaches the known upper obstruction in dimension three throughout the full range $1\le \kappa\le 4$ and therefore gives $$ \epsilon_3(\kappa)=\frac{3}{2\kappa+1} \qquad(1\le\kappa\le4). $$ Beyond this threshold, it yields a closed algebraic lower bound and identifies the precise spectral configuration responsible for the remaining gap. The proof introduces a spectrally resolved continuum-in-opening mechanism for contact geometry. Throughout the contact evolution, the vertex Jacobian retains the negative index of the Hessian and its mean negative curvature—data erased by the classical pointwise reduction—and these variables are optimized only after integration over all openings. This reduces the analysis to a finite family of explicit one-dimensional kernels whose critical exponents recover the sharp planar threshold, identify the exact three-dimensional regime above, and expose a broader principle for extracting second-order regularity from the spectral geometry of contact sets.

We study the initial boundary value problem (IBVP) for the higher order Korteweg–de Vries type equation \[\partial_tu+(-1)^{j+1}\partial_x^{2j+1}u+\frac12\partial_x(u^2)=0,\qquad j\in\mathbb{N},\] on the right half line, subject to the Robin boundary conditions \[(\partial_x+\gamma)\partial_x^{\ell-1}u(t,0)=\varphi_\ell(t),\qquad 1\le\ell\le j,\] where $\gamma\in\mathbb{R}$ is common to all boundary conditions. We prove local well posedness for \[u_0\in H^s(\mathbb{R}^+),\qquad \varphi_\ell\in H^{\frac{s+j-\ell}{2j+1}}(0,T),\qquad -j+\frac14<s<\frac32.\] In particular, the regularity range extends the low regularity theory for the higher order Dirichlet IBVP to Robin boundary conditions. The main ingredient is an explicit unified transform representation for the higher order IBVP with Robin boundary conditions. The Robin hierarchy couples adjacent boundary traces and leads, in each spectral sector, to a nontrivial system for the unknown boundary transforms. Rather than computing the inverse of this system, we resolve precisely the linear combination required by the unified transform representation through Lagrange interpolation at the rotated spectral points. This yields an exact factorization in which the dependence on the Robin parameter is concentrated in the single factor $(k-i\gamma)^{-1}$. Consequently, the only possible Robin pole is $k=i\gamma$, which contributes the residue mode \[e^{-\gamma x+\gamma^{2j+1}t}\] exactly when $j$ is odd and $\gamma>0$. Combining this representation with linear estimates in modified Fourier restriction spaces and higher order KdV bilinear estimates yields the low regularity well posedness result. The representation also recovers the classical KdV formulas with Robin and Neumann boundary data when $j=1$ and is formally consistent with the higher order Dirichlet representation under the reciprocal Robin limit.

In this paper, we prove a global-in-time uniform bound for solutions to a thermo-diffusive system in the whole space with bounded nonnegative initial data. The result allows exponential reaction rates and a range of superexponential rates determined by the diffusivities. This extends the recent results of La, Roquejoffre, and Ryzhik \cite{LRR24} and of Leontev and Ryzhik \cite{LR26} beyond their strictly subexponential range.

We study Maxwell-Stefan diffusion with additive friction coefficients $f_{ij}=g_i+g_j$. In mass fractions, the system isolates the constrained pair-friction block; in mole fractions, it is the classical ideal isothermal/isobaric Maxwell-Stefan system at constant total molar concentration. Additivity makes the constrained pair-friction dissipation species-diagonal; conversely, species-diagonality on one interior barycentric constraint space forces a pair-sum representation. At operator level, the positive constrained relaxation operator is a scalar shift of a compression of $G=diag[g_1,\ldots,g_N]$. Its scalar resolvent yields both an explicit constrained inverse and interlacing spectral roots, which form global real-analytic coordinates on the open simplex and whose differentials are left eigen-covectors. In root coordinates the principal part is diagonal, no self-square gradient term occurs, and scalar comparison yields invariant rectangles and separation from the simplex boundary. For regularity we introduce entropy-stabilized one-sided multi-EPD truncations: Euler-Poisson-Darboux entropies cancel mixed quadratic production, while for $N\ge4$ a truncation-weighted mixing-entropy correction supplies transverse coercivity. Caccioppoli and logarithmic estimates, shrinking, and critical mass yield Hölder continuity up to the Neumann boundary. The mixing entropy also symmetrizes the moment system; frozen conormal estimates give spatial Lipschitz bounds. Together with time Hölder control and short-interval maximal regularity, this yields global strong solvability on bounded $C^{2+\alpha}$ domains with $0<\alpha<1$, for each $N\ge2$, $d\ge2$, and $p>d+2$, for all uniformly positive, compatible initial concentrations in the natural trace class. Solutions become classical for positive times and converge exponentially to equilibrium in relative entropy, $L^2$, and $C^1(\bar\Omega)$.

Let $n\ge3$. We prove a priori interior gradient estimate for the Lagrangian mean curvature equation \[ \sum_{i=1}^n\arctan\lambda_i(D^2u)=\theta(x) \] under the critical and supercritical phase condition $\theta\ge (n-2)\pi/2$, assuming only that the Lipschitz norm of the prescribed phase is bounded.

We introduce and analyse an arbitrary order spatial discontinuous Galerkin (dG) method for the Dean–Kawasaki equation, a highly singular SPDE modelling density fluctuations of $N$ diffusing particles in the regime of large particle number $N \gg 1$. Our starting point is a general procedure for discretising multiplicative, divergence-form noise on finite element spaces whilst preserving its cross-variation structure at the discrete level; the construction is explicit, elementwise, applies to continuous and discontinuous spaces alike, and extends to general mobilities. Using it, we prove weak error estimates of order $O(h^p)$ between fluctuations of the semi-discrete scheme and those of the underlying particle system, together with a correction that is exponentially small in the scaling regime $Nh^d \gg 1$ and arises because the scheme does not preserve positivity. The resulting method is locally and globally conservative and applies on unstructured simplicial meshes. Quantitative numerical experiments support the analysis, and further experiments illustrate the method beyond the scope of the theory: indicator function observables, external and interaction potentials, convection-dominated regimes and reflecting boundary conditions.

Let $N=m+1$ and consider the thin ellipsoid \[ \Omega_\varepsilon=\{(y,x_N)\in\mathbb R^m\times\mathbb R:\ |y|^2+\varepsilon^{-2}x_N^2<1\}. \] We prove that in every sufficiently large dimension, for every sufficiently small $\varepsilon>0$, there exists a smooth positive, strictly increasing, strictly convex, superlinear nonlinearity $f_\varepsilon$ for which the extremal solution in $\Omega_\varepsilon$ is an unbounded $H^1_0$ solution. Combining this result with Dancer's thin-domain regularity theorem for the Gelfand nonlinearity $f(t)=e^t$, we obtain on the same sufficiently thin ellipsoids a bounded Gelfand extremal solution and an unbounded extremal solution for another nonlinearity. Thus, in this two-part sense, Brézis' Open Problem~6.1 is resolved in every sufficiently large dimension.

We initiate the study of nonlinear wave equations on the Witten bubble spacetime (also known as the "bubble of nothing"), which is an $\mathrm{SO}(3,1)\times \mathrm{U}(1)$-symmetric solution to the Einstein vacuum equations in $(4 + 1)$ dimensions. This spacetime was introduced by Witten to model the semiclassical instability of the Kaluza–Klein spacetime $\mathbf{R}^{3+1}\times S^1$ (which is classically stable by work of Huneau–Stingo–Wyatt (arXiv:2307.15267)). We prove a small-data global existence result without symmetry assumptions and an improved decay result for $\mathrm{U}(1)$-symmetric solutions to a class of semilinear equations satisfying a version of the null condition. Key to the proof are novel estimates for the linear wave equation, which has been studied in the physics literature by Bhawal and Viveshwara and in the mathematics literature by Bachelot (arXiv:1601.03682).

We characterize the minimizers of the anisotropic perimeter in the flat torus under an area constraint. We show that, depending on the area, the periodic global minimizer undergoes two phase transitions, changing from a droplet (a rescaled Wulff shape) to a flat band wrapping around the torus in a preferred direction determined by the anisotropy (lamellar phase), and finally to a bubble (the complement of a rescaled Wulff shape). For certain non-symmetric surface tensions, the lamellar phase is absent altogether; for other non regular surface tensions, there exist infinitely many minimizing configurations coexisting with the lamellar phase.

We prove that the Dirichlet $p$-torsion function $u_p$ converges to the distance to the boundary $d$ at the rate $O(1/p)$ in the uniform norm, on every bounded domain and with a constant depending only on the dimension and the diameter. The upper bound comes from a radial barrier with its pole at a boundary point, which is an admissible comparison function for $p>n$ and needs no boundary regularity; the lower bound comes from an inscribed ball. On the ball the error equals $(1+\log n)/p+O(p^{-2})$, so the order $1/p$ cannot be improved in general. Under a uniform exterior ball condition an annular barrier gives $u_p\le K_\Omega^{1/(p-1)}d$ for every $p>1$, with $K_\Omega$ explicit in the dimension, the diameter and the exterior radius; on convex sets the constant is the diameter, and a multiple of $d$ is a supersolution whenever $-\Delta d$ has a positive distributional lower bound. On $C^2$ domains, the same barrier and the classical gradient maximum principle give $\norm{\nabla u_p}_{L^\infty}^{p-1}\le K_\Omega$. When $-\Delta d$ is a measure of finite total variation, which we prove for $C^1$ domains with a uniform exterior ball, the gradients converge in $L^q$ at the rate $O(p^{-1/2})$ for $q\le2$ and $O(p^{-1/q})$ for $q\ge2$; these exponents are not claimed to be sharp. The explicit ball profile shows that the gradients do not converge uniformly.

A criterion for the convexity of a two-dimensional domain is proved in terms of the solution of a boundary value problem for the modified Helmholtz equation.

A time fractional extension of the Hunter Saxton equation is examined, in which the temporal derivative of u_x is replaced by a Caputo derivative of order 0 < alpha <= 1. This modification introduces memory effects into a model traditionally associated with director field dynamics in nematic liquid crystals. By employing a fractional separation of variables strategy together with the exponential spatial profile phi(x) = exp(b+x), which cancels the nonlinear structure exactly, the governing nonlinear partial differential equation is reduced to a fractional relaxation ODE, whose closed form solution is the one-parameter Mittag Leffler function. The exact analytic solution is derived algebraically from the separation process. This appears to be the first exact closed form solution of the Caputo time fractional Hunter Saxton equation. The result is validated symbolically using the MathHandbook computer-algebra system. Quantitative analysis in both the fractional (0 < alpha < 1) and classical (alpha -> 1) limits demonstrates how fractional-order memory slows temporal relaxation relative to the exponential baseline. The solution provides a reliable benchmark for numerical schemes and clarifies how fractional dynamics influence nonlinear wave propagation in orientationally ordered fluids.

Let $(M^n,g_0)$ be a closed smooth Riemannian manifold, with $n\geqslant 3$. We prove short–time existence and uniqueness for the "critical" Ricci–Bourguignon flow \begin{equation} \partial_t g=-2\operatorname{Ric}_g+\frac{R_g}{n-1}g\,. \end{equation} Up to a constant rescaling of time, this is the Schouten flow, since the right–hand side of the equation is $-2(n-2)$ times the Schouten tensor. At the critical parameter, however, the principal symbol of the linearization of the operator that we obtain after the usual "DeTurck modification" still has a zero eigenvalue. We resolve this degeneracy by adjoining an independent scalar unknown $r$, intended to represent the scalar curvature, and considering an "extended" system consisting of a strictly parabolic equation for the metric coupled to a transport–reaction equation for $r$ with a curvature–square forcing. The extended system belongs to the class of nonlinear composite parabolic–hyperbolic systems studied by Vol'pert and Khudyaev~\cite{VolpertKhudyaev1972}. Then, a standard compact–manifold adaptation of their theorem yields a unique smooth solution of the extended system. Finally, the difference $R_g-r$ satisfies a homogeneous linear parabolic equation, so the scalar–curvature constraint "propagates" and the original critical Ricci–Bourguignon flow is recovered. This settles the short–time existence and uniqueness problem at the critical “Schouten” value $\rho=1/(2(n-1))$, which was left open by the previous Ricci–Bourguignon theory in~\cite{CatinoEtAl2017}.

We introduce a nonlinear Poissonian fluctuating Boltzmann equation whose noise encodes both the fluctuations and the path large-deviation rate function of the underlying hard-sphere gas. Unlike the Gaussian-noise equations commonly studied in fluctuating hydrodynamics, the present equation raises a new difficulty: a Poisson random measure produces jumps that may destroy the nonnegativity of the solution. To address this problem, we construct a finite-dimensional coarse-grained jump process and prove its well-posedness and nonnegativity. For each fixed mesh, this process satisfies a good path large-deviation principle. We then fix a regularized velocity cutoff and study the asymptotic behavior of the discrete rate functions as the mesh is refined. On a class of biased regular paths, their limit is the corresponding cutoff Boltzmann large-deviation rate function associated with the hard-sphere gas. This establishes the consistency of the coarse-grained fluctuating Boltzmann model with the underlying particle system at the level of path large deviations.

This paper studies the spectral properties and nonlinear asymptotic stability of a family of finite-time self-similar blow-up solutions to the three-dimensional Keller–Segel system constructed by matching interior and exterior profiles within the framework of matched asymptotic expansions. For every sufficiently large matching index $n$, the full linearized operator around the stationary state $U_n$ in self-similar variables is analyzed on $L^2(\mathbb R^3)$. Sturm zero counting in the radial mode, a wave operator that reduces the nonlocal $l=1$ equation to a local equation, and a Mellin–Newton quadratic form for all $l\ge2$ show that, after the scaling and translation modes and the finitely many genuinely unstable radial modes are removed, the remaining spectrum is separated from the imaginary axis by a positive distance. In addition, the optimal lower bound on the real parts of the spectrum is $1/4$ in every mode $l\ge2$. On the stable subspace, an exponentially decaying semigroup and a modified energy equivalent to the $L^2$ norm are constructed, and the logarithmic asymptotic decay rate of the semigroup norm is proved to equal the stable spectral gap. Finally, modulation equations, $H^2$ energy estimates, control of the scaling derivative, and Brouwer's no-retraction theorem yield nonradial nonlinear asymptotic stability of the corresponding self-similar blow-up solutions after the initial coefficients in the finitely many unstable radial directions have been chosen suitably.

The stochastic Landau–Lifshitz–Gilbert (sLLG) equation is a strongly nonlinear stochastic PDE with a non-convex pointwise constraint arising in the theory of micromagnetics. We analyse a fully discrete, structure-preserving finite element approximation of the sLLG equation with coloured multiplicative Stratonovich noise on a bounded interval. The method utilises continuous piecewise affine finite elements, mass lumping, and midpoint time discretisation to preserve the unit-length constraint exactly at the finite element nodes. Under suitable regularity assumptions on the initial data and the noise, we establish uniform higher-moment stability and develop an error analysis for the scheme. The analysis exploits the geometric structure of the equation and the stochastic midpoint discretisation. For every $\gamma\in(0,\frac12)$, we prove first-order spatial convergence and temporal convergence of order $\gamma$ in the natural discrete energy norm, locally in mean square on events of arbitrarily large probability and, consequently, in probability. To the best of our knowledge, this is the first convergence-rate result for a fully discrete structure-preserving finite element scheme solving the stochastic Landau–Lifshitz–Gilbert equation.

We consider the asymptotic behavior of solutions to inhomogeneous nonlinear Schrödinger equations with non-gauge-invariant nonlinearities. The spatial coefficient in the nonlinear term may have different orders of singularity at the origin and decay at infinity. Under a coercivity condition involving the coefficient and the nonlinearity, we show that no scattering states exist for the equation below a Strauss-type threshold determined by the decay at infinity. Our class includes nonlinearities with a dominant non-oscillatory component. We also prove a complementary small-data scattering result above this threshold, using non-admissible Strichartz estimates in Lorentz spaces adapted to the singular coefficient.

We consider the field enhancement due to the presence of an emitter of the dipole type near two perfectly conducting inclusions with $C^{2,\gamma}$ boundaries in two dimensions. We derive estimates of field enhancement in the narrow region and prove that the field is enhanced by a factor $\varepsilon^{-1/2}$ in the region far from the emitter. We construct a harmonic barrier to control the dipole term instead of using the Kelvin transformation. We also study how the location of the emitter affects the field on the shortest line segment connecting two inclusions.

We derive a two-scale continuum limit for the electrostatic energy of random charge density fields that are stationary and ergodic under lattice translations. Each microscopic cell is charge neutral in every realization, while its dipole moment may fluctuate and have a nonzero mean. Under assumptions on the stationary microscopic potential and the finite-volume fields, the local and nonlocal energies converge almost surely to deterministic limits. The local limit consists of the ensemble-averaged microscopic Coulomb energy and a cell-depolarization term, whereas the effective polarization determines the nonlocal limit. The cell-depolarization terms cancel in the total energy, which consists of the ensemble-averaged microscopic cell energy and the macroscopic electrostatic field energy. The deterministic specialization recovers the periodic two-scale limit. For an independent-cell random-displacement model, we verify the assumptions of the continuum-limit theorem and show that centered microscopic fluctuations can change the local energy without changing the effective polarization or the nonlocal energy.

All recent advances \cite{DRS}, \cite{S-Rita1}, \cite{S-Rita2}, \cite{Millet}, \cite{MaS}, \cite{MaS2} on the boundedness and decay for wave equations on subextremal Kerr spacetimes $\KK(a,m)$, with non-small angular rotation, rely in an essential way on Whiting's groundbreaking discovery of a powerful integro-differential transformation with the help of which he ruled out exponentially growing modes in \cite{W} (see also \cite{Yacov}, \cite{AMPW} and \cite{Rita} for further developments). We give a purely physical space, generalized version, of the transformation which takes general solutions of {spin-$\sk$} wave equations in $\KK=\KK(a,m)$ to solutions of a secondary wave equation on a new metric manifold $\KKt=\KKt(a,m)$ with two stationary, asymptotically flat, ends. Moreover, for a certain subdomain $\DDt\subset\DDt_-$, which can be identified as the exterior of a black hole region, the new metric $\gt$ of $\KKt(a, m)$ is Lorentzian and the time translation $\T $ is timelike. Using the geometric properties of the two ends one can then deduce, by classical asymptotic Fourier analysis techniques, the desired bound for the flux of $\psi$ at the future event horizon. This estimate played a crucial role in our companion paper \cite{He-K1}.

This paper is concerned with the Cauchy problem for the one-dimensional bipolar Euler-Poisson equations with space-dependent damping, which arise in the hydrodynamic model of semiconductors. Notably, the two pressure functions are allowed to be different, and the doping profile is nonzero. By means of the time-weighted energy method with carefully chosen weights, we prove that if the initial perturbation around a constant steady-state is small enough, the Cauchy problem admits a unique global smooth solution, which converges to the constant steady-state at algebraic decay rates. We show that the decay rates coincide with those established for the constant-coefficients damping case, thereby overcoming the influence of spatially dependent damping on the decay rates of the solutions. A key feature is that no smallness of the space-dependent damping coefficients is required.

We analyze an inverse problem for determining the piston speed and the associated flow field from a prescribed leading shock and the initial data in a shock tube. The gas flow is described by the isentropic Euler equations (i.e., the $p$-system), while the trajectory of the leading shock is prescribed as a given Lipschitz curve. Under an Oleĭnik-type entropy condition on the leading shock, we develop a modified wavefront tracking scheme to construct the flow field behind the shock. This construction enables us to determine the corresponding piston speed and the associated flow field.

Let $L^2_m(\mathbb{R}^2)$ be the space of square-integrable functions on $\mathbb{R}^2$ with $m$-fold rotational symmetry. Let $\overline{\omega}(r,\theta) = r^{-\alpha}f(\theta)$ be a homogeneous steady state of the two-dimensional incompressible Euler equations with $(m\cdot l)$-fold rotational symmetry. If $f(\theta)$ is a constant function we say that $\overline{\omega}$ is a radial power-law vortex. We prove that the incompressible Euler equations in vorticity form, linearized around any homogeneous steady state $\overline{\omega}$ that is not a radial power-law vortex, are ill-posed on $L^2_m(\mathbb{R}^2)$ for any $m\geq 2$, any $l\geq 1$ and $\alpha\in (0,1)$.

In 1966, McKean conjectured that the entropy production of Kac's caricature of a Maxwellian gas is nonincreasing in time \cite{McKean66}. We disprove this conjecture for the standard uniform-angle Kac equation. Our counterexamples are smooth, strictly positive, even mixtures of two Gaussians at different temperatures. We also show that every entropy derivative beyond the first violates its expected sign at some time along the same solution. The initial densities can be chosen with unit mass and second moment. mass and second moment.