Papers

We prove scattering in $L^2(\mathbb{R}^2)$ for real-valued solutions of the Kadomtsev–Petviashvili II (KP-II) equation with localized initial data. No smallness assumption is made, no zero mean condition is imposed, and the admissible data include all Schwartz functions. The proof uses the inverse scattering transform attached to the heat operator. It rests on three ingredients: an a priori bound for the scalar $\bar\partial$ problem which depends on the size of the scattering coefficient but not on its oscillating phase, a nonlinear Plancherel identity proved by analytic continuation from small data, and an estimate showing that the nonlinear part of the reconstruction is negligible on the propagation cones of free waves. To the best of our knowledge, this is the first result establishing scattering for the KP-II equation for every real-valued Schwartz initial datum.

This work is concerned with the focusing quintic nonlinear Schrödinger equation on a star graph with two edges, with a delta interaction of arbitrary strength at the vertex. We construct a non-radial, minimal mass, finite time blow-up solution, whose singularity forms at an arbitrarily prescribed interior point of one edge. Our approach is based on a suitable cut-off of the standard pseudoconformal blow-up solution of the focusing quintic NLS on the line, inspired by Ogawa-Tsutsumi and Burq-Gérard-Tzvetkov. A noteworthy feature of our construction is the persistence of the domain of the Hamiltonian under the flow, obtained by the persistence of regularity method of Kato. We also identify a two-parameter family of distinct blow-up profiles, which should play a central role in the classification of non-radial minimal mass blow-up solutions on the star graph. Our construction is generic and can be extended to a star graph with an arbitrary number of edges. It can certainly also be adapted to other types of metric graphs.

We study the gauge of the travel time tomography problem for elliptically anisotropic planar media. Under this assumption the respective slowness surfaces consist of an inner ellipse and an outer circle. Hence, the singularities of seismic waves propagate on geodesics given by two Riemannian metrics of which the one related to shear waves is conformally Euclidean. We show that if both of these metrics are lens rigid then, up to boundary fixing diffeomorphisms, each elliptically anisotropic stiffness tensor is uniquely determined by the lens data whenever it has at least one isotropic point. If none of those points exists, then there are exactly two elliptically anisotropic stiffness tensors with the same lens data. Clearly, not all boundary fixing diffeomorphisms preserve the imposed structure of the slowness surface. We show that the set of well-behaved diffeomorphisms is never a subgroup of all boundary fixing diffeomorphisms. Our proofs are based on an interplay of algebra and geometry, and the key of the proofs is a characterization of the elliptic anisotropy via several polynomial conditions.

By establishing a weighted Lojasiewicz-Simon inequality for maps which are close to a fixed harmonic map modulo scaling, we prove Topping's conjecture on continuity of the body map in the case of finite time singularities at which only isolated bubbles form.

We obtain sharp isoperimetric-type upper bounds for Steklov eigenvalues on warped products whose bases are compact Riemannian manifolds with boundary and whose fibres are closed Riemannian manifolds. These bounds involve the volume of the warped product and of its boundary, as well as the eigenvalues of the Laplace operator on the fibre and the Steklov eigenvalues of the base. After separation of variables, these bounds are obtained by comparison with auxiliary Helmholtz-Steklov problems on the base involving weighted Laplace operators naturally induced by the warping function. We also study the effect of various natural geometric constraints, such as the warping function being identically equal to one on the boundary. The proofs make use of various spectral approximation methods, including homogenisation on manifolds when the fibres are one-dimensional.

We consider the Helmholtz equation in a Dirichlet angular sector (which may be locally perturbed) covered on one side by a thin homogeneous layer. First, we propose an asymptotic expansion of the solution at any order and for any angle when the thickness of the layer tends to 0, using a matched asymptotic expansion. This consists in introducing different asymptotic expansions in three subdomains – the vicinity of the corner, the layer and the rest of the domain – that are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have complex singular behaviors. We overcome this difficulty using a novel system of algebraic tools, including formal series and operators on these series. Moreover, we justify the asymptotic expansion with error estimates and provide details on the form of the expansion depending on the angle. We numerically validate this expansion up to order 5. This requires computing the behavior at infinity of some corner profiles. To do this, we introduce a Galerkin-type method enriched by complements at infinity. These complements are known functions that appear in the behavior at infinity of the corner profiles with coefficients that become unknowns of the system. Finally, we propose effective models of order 3, resp. 5 where, far from the corner, the layer is replaced with an effective Robin, resp. Ventcel, boundary condition and, near the corner, an artificial boundary is introduced and a Dirichlet-to-Neumann (DtN) type condition is imposed. The DtN map relies on a modal decomposition of the truncated asymptotic expansion at order 3, resp. 5. The modes take into account the singular behavior of the expansion terms, the corner profiles and some algebraic matching coefficients. We prove the well-posedness of our effective models, as well as error estimates, and provide numerical results.

We consider the Burgers-type stochastic PDE \[ du = \theta\,\partial_x^2 u\,dt + f(u)\,dt + g(u)\,\partial_x u\,dt + \sigma\,dW(t), \] driven by space–time white noise on the one-dimensional torus, where \(\theta>0\) is the diffusivity parameter and \(\sigma>0\) the noise intensity. The transport term \(g(u)\,\partial_x u\) admits no classical interpretation, and we work with the mild solutions defined via rough paths following~\cite{Hairer2011}, for which the solution decomposes as \(u=\psi+R\) with \(\psi\) a stationary centred Gaussian process solving a damped stochastic heat equation and \(R\) a remainder. We first show that, away from the initial time, \(R\) is Lipschitz in space and Hölder in time of order strictly greater than \(1/2\), hence strictly smoother than \(\psi\) in both variables. We then compute exactly the spatial quadratic variation and the temporal quartic variation of \(\psi\), almost surely and not merely in \(L^{2}(\Omega)\), and prove that the corresponding variations of \(u\) are unaffected by the remainder. Combined with an exact time-scaling identity, this separation of regularity scales yields strongly consistent pathwise estimators of \(\theta\). To our knowledge this is the first inference procedure for an SPDE whose solution is defined only pathwise, as a measurable function of a rough path, rather than as an Itô process.

An inverse spectral problem for a $(1+1)$-dimensional wave equation with wave speed $c(x)$ is considered. The spectral data consist of the eigenvalues of the Neumann–Dirichlet problem for the operator $-c^2(x)\partial_x^2$ together with their derivatives with respect to a Robin parameter at the boundary. We propose a two-step recovery procedure. In the first step, the Neumann-to-Dirichlet map $\Lambda$ is reconstructed from the spectral data using the results of arXiv/0406582. In the second step, the reconstructed $\Lambda$ is used within the regularization strategy for the boundary control method presented in arXiv:1509.04478. As an improvement, the regularization of the control problems employs a weighted combination of $L^2$ and $H^1$ penalty terms, and we prove a uniform convergence result for the family of optimal controls, thereby allowing the regularization parameters to be tuned simultaneously. We also prove that the forward spectral map, which assigns to a wave speed its spectral data, is Lipschitz continuous, and that the inverse spectral map underlying the reconstruction procedure is Hölder continuous on a subclass of wave speeds. The method is implemented numerically and validated on both smooth and piecewise smooth wave speed profiles. Numerical experiments demonstrate that the recovery of the wave speed exhibits high accuracy and robustness.

We study the two-dimensional compressible Navier-Stokes-Poisson system in a general simply connected bounded domain with Navier-slip boundary conditions and a positive, possibly non-flat, doping profile. We establish the global existence of a unique strong solution and of a global weak solution for arbitrarily large initial data allowing vacuum when the shear viscosity is a positive constant and the bulk one proportional to a power of the density with the power bigger than 4/3. The main difficulties arise from the interaction between the Poisson force and the general boundary geometry. To overcome them, we introduce a modified material derivative and derive quantitative $L^p$ estimates for the density by means of a Neumann duality argument together with a harmonic correction generated by the Navier-slip boundary condition. We further obtain a pointwise representation of the effective viscous flux by pulling back the Neumann Green function on the unit disk through a conformal mapping. The resulting commutator estimates, combined with elliptic estimates for the Poisson equation, yield a uniform-in-time upper bound for the density and allow us to close the higher-order a priori estimates. Finally, the global weak solution is constructed through a charge-preserving approximation and compactness argument.

We establish endpoint partial regularity for suitable weak solutions to the three-dimensional chemotaxis–Navier–Stokes system with constant sensitivity and linear consumption in a half-space under no-slip and no-flux boundary conditions. The full space–time singular set, including its boundary part, has vanishing one-dimensional parabolic Hausdorff measure. This follows from a Caffarelli–Kohn–Nirenberg type criterion at the natural $r^{-1}$ dissipation scaling, valid at interior and boundary points without additional pressure smallness. The key estimate combines weighted mass control with the localized entropy–energy inequality to close the iteration without a separate $n\log n$ quantity and accounts for the mass term generated by rescaling. Boundary-compatible pressure and caloric decompositions yield local Hölder regularity. We also prove a lower-order criterion requiring smallness only of the velocity, signal gradient, and pressure oscillation, with a threshold depending on an arbitrary finite density mass bound. We construct global suitable weak solutions under natural entropy and energy assumptions without smallness restrictions on the initial data. The potential is time-independent with bounded gradient; boundedness of the potential itself is assumed only for the large-time results. If its gradient is also Hölder continuous, a solution can be selected to become bounded and classical throughout the closed half-space after a finite waiting time. For the same solution, we obtain decay and Neumann Gaussian asymptotics for both scalar components without initial spatial moment assumptions. The profile coefficients are the conserved bacterial mass and the limiting signal mass, respectively, and the latter is strictly positive whenever the initial signal is nonzero.

In an earlier study of the Laplacian eigenvalues with strongly attractive Robin boundary conditions in polygons it was found that if each corner is either concave or non-resonant, then the asymptotic behavior of the individual eigenvalues is determined by an effective one-dimensional operator on the boundary with Dirichlet boundary conditions at the corners. The non-resonance condition was formulated in terms of an eigenvalue bound on truncated sectors, and it was only shown to be satisfied by the right and obtuse opening angles. In the present work, we determine the full range of non-resonant angles: it turns out that the non-resonance condition fails only for the opening angles $\frac{\pi}{2n+1}$ with $n=1,2,3,\dots$, and these critical values are equivalently characterized in terms of the jumps of the eigenvalue counting function and the existence of bounded generalized eigenfunctions. This resolves the numerically supported conjecture stated by Dauge and Lafranche.

We prove that the local Dirichlet-to-Neumann map of $\Delta_g+q$ determines a compact connected real-analytic Riemannian manifold of dimension at least three and a real-analytic real-valued potential, up to a diffeomorphism fixing the observed boundary and the conformal gauge of the equation. The result is new even when \(q\) is constant, i.e. for the fixed-frequency (Helmholtz) equation \(\Delta_g u+\lambda u=0\). The measurements are restricted to an arbitrary nonempty open boundary set $\Gamma$. The proof develops conformal Poisson embeddings method and uses coordinates given by quotients of solutions.

We study the long-time behaviour of multi-dimensional first-order symmetrizable quasilinear hyperbolic systems with nonlinear feedback boundary conditions. Assuming strict interior and boundary Lyapunov inequalities for the linearized system and non-characteristic boundaries, we prove global existence for small initial data and exponential H^s decay, with s>d/2+1 an integer. We consider both bounded smooth domains and flat periodic cylinders, and illustrate the results with physical examples and numerical simulations.

The parabolic-elliptic Keller-Segel system \[ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - \mu + u, \qquad \mu:=\frac{1}{|\Omega|} \int_\Omega u, \end{array} \right. \qquad \qquad (\star) \] is considered under homogeneous Neumann boundary conditions in the ball $\Omega=B_R(0)\subset {\mathbb R}^n$, where $n\ge 2$ and $R>0$. The main results identify a mass level $m_\star(n,R)>0$ as critical in the sense that $\bullet$ whenever $0<m<m_\star(n,R)$, the constant equilibrium of ($\star$) given by $u\equiv v\equiv \frac{m}{|B_R|}$ is locally asymptotically stable with respect to the norm in $L^\infty(B_R)$, but that $\bullet$ if $m>m_\star(n,R)$, then any solution of ($\star$) emanating from nonnegative and nonconstant continuous radial initial data $u_0$ with $\int_{B_R} u_0=m$ and $\frac{1}{|B_r|} \int_{B_r} u_0 \ge \frac{m}{|B_R|}$ for all $r\in (0,R)$ blows up in finite time. The value of $m_\star(n,R)$ can be determined in an essentially explicit manner in dependence on principal zeroes of appropriate Bessel functions of the first kind; in particular, $m_\star(2,R)\approx 14.68197 \cdot \pi$ for all $R>0$. Accordingly, as a supplement to known mass-dependent blow-up dichotomies these findings reveal a genuinely secondary critical mass phenomenon for ($\star$) especially also in its two-dimensional version in which no precedent of this form seems known.

In this paper, we study Hölder regularity of local weak super-solutions to the parabolic equation associated with fractional $p$-Laplacian type operators $$ \partial_t u(t,x)-\text{p.v.}\int_{\R^d}|u(t,y)-u(t,x)|^{p-2}(u(t,y)-u(t,x))J(t;x,y)\,dy=0,\quad (t,x)\in \R\times \R^d, $$ where $\text{p.v.}$ denotes the principal value, $p\in(1,\infty)$ and the kernel $J(t;x,y)$ is comparable (uniformly with respect to $(t;x,y)\in\R\times\R^d\times\R^d$) to the fractional $p$-Laplacian kernel $|x-y|^{-d-sp}$ with $s\in(0,1)$. Our contribution extends the fundamental work of Kassmann and Weidner to the nonlinear case $p\ne 2$. In particular, we provide an explicit nonlocal parabolic tail condition in terms of an $L^{1+\varepsilon}$-norm in the time variable (for a suitable $\varepsilon>0$) for all $p\in(1,\infty)$, under which Hölder regularity holds. Our proof is based on the method of expansion of positivity, which is the key ingredient in the reduction-of-oscillation argument.

We study the asymptotic behavior of eigenvalues of nonlocal convolution-type operators $$L_\varepsilon\rho=-\frac{1}{\varepsilon^{d}}\int_\Omega J\left({\frac{|x-y|^2}{\varepsilon^2}}\right)\rho(y)dy+a(x)\rho(x),$$ (with a continuous sufficiently fast decaying at infinity function $J\geq 0$) when the scaling parameter $\varepsilon>0$ tends to zero. Assuming that the minimum of the function $a(x)$ is attained at a point $x_0\in \overline{\Omega}\subset \mathbb{R}^d$ and is non-degenerate, we establish several terms in the asymptotic expansion of the first eigenvalues $\lambda_\varepsilon^{(k)}$ of $L_\varepsilon$. If $x_0$ is an interior point of the domain $\Omega$, we show that $\lambda_\varepsilon^{(k)}=a(x_0)-1+\varepsilon\mu^{(k)}+\overline{o}(\varepsilon)$, where $\mu^{(k)}$ are eigenvalues of a harmonic oscillator (obtained by the blow up analysis), the corresponding eigenfunctions of $\mathcal{L}_\varepsilon$ are concentrated near $x_0$ on the scale $\varepsilon^{1/2}$. If $x_0\in\partial \Omega$ then the asymptotic expansion of the eigenvalues of $L_\varepsilon$ has the form $\lambda_\varepsilon^{(k)}=a(x_0)-1+\varepsilon^{2/3}\Lambda_0+\varepsilon(\mu^{(k)}+\alpha p_0)+\overline{o}(\varepsilon)$. The corresponding eigenfunctions are represented as the product of a function depending on the distance to $\partial\Omega$ and concentrated on the scale $\varepsilon^{2/3}$, and functions localized on the scale $\varepsilon^{1/2}$ and asymptotically independent of the distance to $\partial\Omega$. The latter functions converge (after rescaling) to eigenfunctions of a harmonic oscillator (in tangential variables), $\mu^{(k)}$ being their eigenvalues. The former function is given, to the leading term, by the scaled Airy function, and $\Lambda_0$, $\alpha$ and $p_0$ depend on $J(|x|^2)$ and $a(x)$.

In order to control epidemics of mosquito-borne diseases such as dengue, for which there is still no vaccine available, several strategies aim to directly target the mosquito population. One technique involves introducing a symbiotic bacterium called Wolbachia into the population, which blocks the transmission of pathogens. A mathematical model of the introduction of this bacterium and the replacement of the mosquito population with a population carrying this bacterium has been developed, proving that in a homogeneous environment, there is spatial propagation of the replacement. The objective of this paper is to study the blocking of this propagation in cases where the environment is not homogeneous. In particular, we are considering scenarios where the environmental carrying capacity varies significantly across different regions of the spatial domain. To achieve this objective, we first reduce the original 2x2 reaction-diffusion system to a single closed equation, providing a simplified framework for studying the dynamics of Wolbachia propagation. We then investigate the blocking of the propagation in the population replacement process, focusing on the effects of spatial variations in the environmental carrying capacity. Numerical simulations have been performed to validate the theoretical results.

Consider \begin{equation*}\ \left\{ \begin{aligned} -\mathcal_{\lambda,\Lambda}^{+}(D^{2}u)&=\mu u+a(x)u^{p}\quad\text{in}~\Omega,\\ u&=0\quad\text{on}~~\partial\Omega, \end{aligned} \right. \end{equation*} where $ \ Omega$ is a smooth bounded domain, $a$ is a continuous function which is allowed to change sign. Moreover, the set of zeros of $a$ may have a non-empty interior. The main objective of this article is to establish a uniform bound on the set of positive solutions of the above problem. This approach requires a suitable Liouville-type theorem. We first prove the Liouville-type theorem for $-\mathcal_{\lambda,\Lambda}^{+}(D^{2}v)=h(x_{1})f(v),$ where \(h\) is a strictly increasing one-dimensional weight and \(f\) is positive and nondecreasing.

Let $\Omega$ ___ C n be a bounded hyperconvex domain and $\Phi$\,: $\Omega$ $\rightarrow$ R a maximal plurisubharmonic function in $\Omega$ (possibly unbounded). We introduce a new class E 1 $\Phi$ ($\Omega$) of plurisubharmonic functions with boundary values equal to $\Phi$ in the sense of Cegrell and finite $\Phi$-energy on $\Omega$. Then given a Borel measure $\mu$ on $\Omega$, we apply a variational approach to prove that $\mu$ is the Monge-Ampère measure of a function u $\in$ E 1 $\Phi$ ($\Omega$) if and only if it satisfies the integrability condition We also establish a quantitative version of this condition extending previous results of Cegrell and others. Using a logarithmic transformation, we obtain analogous results for the real Monge-Ampère equations with (unbounded) boundary data.

These notes summarize the content of a one-week course given by Luis Vega on the binormal flow model applied to describe the dynamics of three-dimensional vortex filaments in incompressible fluids following the Euler equations. This topic touches connections between the binormal flow, the Schrödinger map and the 1d cubic nonlinear Schrödinger equation, providing a rich casuistry linked to the physical and numerical phenomenology. This course was part of the $2^{\text{nd}}$ Advanced Summer School held at the Institut d'Etudes Scientifiques de Cargèse (IESC), at Cargèse (Corsica), from $20^{\text{th}}$ to $30^{\text{th}}$ June 2023. These notes have been written and completed by two of its attendees, Elena Danesi and Pablo Merino.

We study the recovery of a three-dimensional Riemannian manifold from the areas of minimal surfaces. We show that areas of minimal surfaces determine the metric on manifolds foliated by minimal surfaces, up to boundary-fixing isometry. This resolves the conjecture on boundary rigidity for least-area data in an exterior formulation, even without topological restrictions on the minimal surfaces. We also prove global rigidity for Einstein metrics on compact three-manifolds with boundary, assuming $\(\pi_1(M,\partial M)=0\)$ and a non-trapping condition. The Einstein result includes the

We consider second-order strictly hyperbolic operators whose coefficients are Lipschitz continuous in space and may oscillate near the initial time. Their first two time derivatives are bounded by the corresponding powers of $\frac{1}{t}(\log\frac{1}{t})^\alpha$ with $0\leqslant\alpha<1$, and the first time derivative satisfies the corresponding spatial Lipschitz bound. By combining Tarama's energy with frequency-dependent weights, we prove an energy estimate without loss for $\alpha=0$ and with a stretched logarithmic upper bound for the loss when $0<\alpha<1$, which implies an arbitrarily small loss in the Sobolev scale. These estimates extend the analysis of Colombini, Del Santo and Fanelli [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 26 (2025)] for spatially Lipschitz coefficients to time-derivative bounds with an additional logarithmic factor.

We consider the Navier-Stokes equations in a smooth bounded domain $D\subset \mathbb R^3$. It has been proved in arXiv:2609.08284 that these equations have a unique solution - globally defined in time - when a superlinear noise is introduced and the initial velocity belongs to the space $V$ of divergence-free $H^1_0(D;\mathbb R^3)$-vector fields. Here we show that the same result holds when there is also an additive noise and a deterministic forcing term. Moreover, when the deterministic forcing term is time-independent we prove the existence of at least one invariant measure for this stochastic equation. This is obtained by the Krylov-Bogoliubov method, based on tightness and Feller property in $V$. Finally, we show that the set of invariant measures contains one ergodic invariant measure.

We consider the Cauchy problem on the real line for a one-dimensional transverse-field isentropic non-resistive magnetohydrodynamic system with viscosity $\mu(\rho)=\alpha\rho^\delta$, $0<\delta\leq1$, and vacuum at spatial infinity. The viscosity therefore vanishes precisely where the density vanishes. We establish two global large-data theorems, one for the sublinear range $0<\delta<1$ and one for the linear case $\delta=1$, in a nontrivial invariant class of magnetic fields. The class is described by the frozen-in flux $h=b/\rho$ and its mass derivative $k=\rho^{-1}h_x$. The key identity $[\partial_t+u\partial_x,\rho^{-1}\partial_x]=0$ transports both $h$ and $k$, removes all negative powers of the density from the Lorentz force, and propagates the magnetic regularity. A modified effective-velocity entropy contains an additional coercive magnetic term, while its only error is controlled by $k$. Magnetic pressure also strengthens the density upper bound. These estimates close the large-data continuation argument in the ranges $0<\delta<1$, $\gamma\geq\delta+\tfrac12$, and $\delta=1$, $\gamma>\tfrac32$. As a result, we extend the work of Cao, Li, and Zhu (SIAM J. Math. Anal., 2022) from the Navier-Stokes equations to the MHD equations.

For diffusions whose coefficients are smooth, with at most linear growth and bounded derivatives of higher order, we establish explicit pointwise bounds on the spatial derivatives of the classical solution to the Cauchy problem with polynomially growing initial data. We also obtain explicit growth estimates for the derivatives of the SDE with respect to the initial data. The proofs combine Lyapunov functions and Markov semigroups on weighted spaces with Kunita's theory of the differentiable dependence of SDE solutions on the initial data.

In this paper, we study the relation between different notions of monotonicity for isotropic hyperelastic compressible and incompressible materials. We study how to define the monotonicity concept on the manifold $\rm{Sym}^{++}(n)\cap \rm{SL}(n)$ of positive definite symmetric tensors with unit determinant and the corresponding spectral manifold $\mathcal S=\left\{(\lambda_1,\dotsc,\lambda_n)\in \mathbb{R}^n_+\,\middle|\, \lambda_1\lambda_2\cdots\lambda_n=1\right\}$, and give a precise characterisation of monotonicity in the case of hyperelastic materials. In particular, we consider the monotonicity properties of the Biot stress tensor $T_{\rm Biot}(U)$, the engineering stress, or the load tensor.

In this paper, we prove global pointwise estimates on the decreasing rearrangement \(u^{\ast}\) and \([H(\nabla u)]^{\ast}\) of SOLAs \(u\) of an elliptic Poisson equation driven by the anisotropic \(p\)-Laplace operator \(\Delta_p^H\) for given \(L^1\)-data and equipped with either homogeneous Dirichlet or Neumann boundary conditions. Here, we assume that the dimension \(n\ge 2\), \(1<p<n\) and \(H\) is norm on \(\R^n\). In the degenerate range \(2 \leq p <n\), this gives an anisotropic extension and a full-interval completion of the rearrangement theory of Cianchi-Maz'ya [JEMS, 2014]. In particular, we obtain a new rearrangement estimate for gradients by decomposing the data for singular range \(1<p<2\), where the nonlinear interpolation method aren't available. Moreover, the endpoint weak-type gradient estimate is obtained for all \(1<p<n\). In the range \(p_*(n)<p<n\) with \(p_*(n):=\frac{3n-2}{2n-1}\), we prove a stability estimate for differences of gradients. These results extends the known ones in the literature to the singular range \(p_*(n)<p<2\).

We classify all nonconstant smooth harmonic maps $\mathbb{S}^2\to \mathbb{CP}^n$ and $\mathbb{S}^3\to \mathbb{CP}^n$ whose 0-homogeneous extensions are minimizing: they are, respectively, isometric embeddings or holomorphic isometric embeddings composed with a Hopf map.

We investigate the time-asymptotic stability of composite waves consisting of a non-classical Liu shock and a rarefaction wave for non-convex viscoelastic system. By choosing delicate time-dependent shift functions to the viscous shock and suitable weight functions for the relative entropy, we succeed in proving the time-asymptotic stability of the composite of a viscous Liu shock and a rarefaction wave from two different characteristic families for the one-dimensional viscoelastic system with a cubic non-convex stress-strain relation by $a$-contraction method. Here the non-classical Liu shock can be either degenerate or non-degenerate, which can be seen as the system version of the scalar Oleinik shock. Note that both the shift and the weight functions are chosen quite differently in the two cases of nearly degenerate and fully non-degenerate. The paper represents the first time-asymptotic stability result for the composite waves consisting of both non-classical Liu shock and rarefaction wave from two different families for the non-convex physical system.

We recall several derivations of temperature-dependent Cahn-Hilliard equations. The resulting equations are systems of mixed order. For a suitable subclass of equations we discuss its local-in-time well-posedness based on $L^p$-maximal regularity theory. We also discuss stability of equilibria.