Commun Pure Appl Math Research Articles

Existence and Stability of Infinite Time Blow-Up in the Keller–Segel System

Perhaps the most classical diffusion model for chemotaxis is the Keller–Segel system We consider the critical mass case ∫R2u0(x)dx=8π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\int _{{\\mathbb {R}}^2} u_0(x)\\, \ extrm{d}x = 8\\pi $$\\end{document}, which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function u0∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0^*$$\\end{document} with mass 8π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi $$\\end{document} such that for any initial condition u0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0$$\\end{document} sufficiently close to u0∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0^*$$\\end{document} and mass 8π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8\\pi $$\\end{document}, the solution u(x, t) of (∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}) is globally defined and blows-up in infinite time. As t→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\rightarrow +\\infty $$\\end{document} it has the approximate profile u(x,t)≈1λ2(t)Ux-ξ(t)λ(t),U(y)=8(1+|y|2)2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u(x,t) \\approx \\frac{1}{\\lambda ^2(t)} U\\left( \\frac{x-\\xi (t)}{\\lambda (t)} \\right) , \\quad U(y)= \\frac{8}{(1+|y|^2)^2}, \\end{aligned}$$\\end{document}where λ(t)≈clogt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda (t) \\approx \\frac{c}{\\sqrt{\\log t}}$$\\end{document}, ξ(t)→q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi (t)\\rightarrow q$$\\end{document} for some c>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c>0$$\\end{document} and q∈R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\in {\\mathbb {R}}^2$$\\end{document}. This result affirmatively answers the nonradial stability conjecture raised in Ghoul and Masmoudi (Commun Pure Appl Math 71:1957–2015, 2018).

Energy scaling laws for microstructures: from helimagnets to martensites

We consider scalar-valued variational models for pattern formation in helimagnetic compounds and in shape memory alloys. Precisely, we consider a non-convex multi-well bulk energy term on the unit square, which favors four gradients (±α,±β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\\pm }\\,\\alpha ,{\\pm }\\,\\beta )$$\\end{document}, regularized by a singular perturbation in terms of the total variation of the second derivative. We derive scaling laws for the minimal energy in the case of incompatible affine boundary conditions in terms of the singular perturbation parameter as well as the ratio α/β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha /\\beta $$\\end{document} and the incompatibility of the boundary condition. We discuss how well-studied models for martensitic microstructures in shape-memory alloys arise as a limiting case, and relations between the different models in terms of scaling laws. In particular, we show that scaling regimes arise in which an interpolation between the rather different branching-type constructions in the spirit of Kohn and Müller (Commun Pure Appl Math 47(4):405–435, 1994) and Ginster and Zwicknagl (J Nonlinear Sci 33:20, 2023), respectively, occurs. A particular technical difficulty in the lower bounds arises from the fact that the energy scalings involve various logarithmic terms that we capture in matching upper and lower scaling bounds.

Dispersive Hydrodynamics of Soliton Condensates for the Korteweg–de Vries Equation

We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg–de Vries (KdV) equation in the special “condensate” limit. We prove that in this limit the integro-differential kinetic equation for the spectral density of states reduces to the N-phase KdV–Whitham modulation equations derived by Flaschka et al. (Commun Pure Appl Math 33(6):739–784, 1980) and Lax and Levermore (Commun Pure Appl Math 36(5):571–593, 1983). We consider Riemann problems for soliton condensates and construct explicit solutions of the kinetic equation describing generalized rarefaction and dispersive shock waves. We then present numerical results for “diluted” soliton condensates exhibiting rich incoherent behaviors associated with integrable turbulence.

One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams

We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Gamma -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Gamma -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004).

On a Fractional Nirenberg Problem Involving the Square Root of the Laplacian on $${\mathbb {S}}^{3}$$

In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $$n=3,$$ $$\sigma =1/2,$$ when the prescribing $$\sigma $$ -curvature function satisfies the $$(n-2\sigma )$$ -flatness condition near its critical points. The compactness results are new and optimal. In addition, we obtain a degree-counting formula of all solutions. From our results, we can know where blow up occur. Moreover, for any finite distinct points, the sequence of solutions that blow up precisely at these points can be constructed. We extend the results of Li (Commun Pure Appl Math 49:541–597, 1996) from the local problem to nonlocal cases.

Phase separating solutions for two component systems in general planar domains

In this paper we consider a two component system of coupled non linear Schrödinger equations modeling the phase separation in the binary mixture of Bose–Einstein condensates and other related problems. Assuming the existence of solutions in the limit of large interspecies scattering length $$\beta $$ the system reduces to a couple of scalar problems on subdomains of pure phases (Noris et al. in Commun Pure Appl Math 63:267–302, 2010). Here we show that given a solution to the limiting problem under some additional non degeneracy assumptions there exists a family of solutions parametrized by $$\beta \gg 1$$ .

Rates of Convergence to Non-degenerate Asymptotic Profiles for Fast Diffusion via Energy Methods

This paper is concerned with a quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy–Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents. More precisely, rates of convergence to non-degenerate asymptotic profiles are revealed via an energy method. The sharp rate of convergence to positive ones was recently discussed by Bonforte and Figalli (Commun Pure Appl Math 74:744-789, 2021) based on an entropy method. An alternative proof for their result is also provided. Furthermore, the dynamics of fast diffusion flows with changing signs is discussed more specifically under concrete settings; in particular, exponential stability of some sign-changing asymptotic profiles is proved in dumbbell domains for initial data with certain symmetry.

A Rigidity Result for the Robin Torsion Problem

Let Omega subset mathbb {R}^2 be an open, bounded and Lipschitz set. We consider the torsion problem for the Laplace operator associated to Omega with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison, proved in Alvino et al. (Commun Pure Appl Math 76:585–603, 2023).. We prove that the equality is achieved only if Omega is a disk and the torsion function u is radial.

Learning Invariance Preserving Moment Closure Model for Boltzmann–BGK Equation

As one of the main governing equations in kinetic theory, the Boltzmann equation is widely utilized in aerospace, microscopic flow, etc. Its high-resolution simulation is crucial in these related areas. However, due to the high dimensionality of the Boltzmann equation, high-resolution simulations are often difficult to achieve numerically. The moment method which was first proposed in Grad (Commun Pure Appl Math 2(4):331–407, 1949) is among the popular numerical methods to achieve efficient high-resolution simulations. We can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation, which effectively reduces the dimensionality of the problem. However, one of the main challenges is that it leads to an unclosed moment system, and closure is needed to obtain a closed moment system. It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory. Other than the traditional human designs of closures, the machine learning-based approach has attracted much attention lately in Han et al. (Proc Natl Acad Sci USA 116(44):21983–21991, 2019) and Huang et al. (J Non-Equilib Thermodyn 46(4):355–370, 2021). In this work, we propose a machine learning-based method to derive a moment closure model for the Boltzmann–BGK equation. In particular, the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances, i.e., the Galilean invariance, reflecting invariance, and scaling invariance, inherited from the original Boltzmann–BGK equation and playing an important role in the correct simulation of the Boltzmann equation. Numerical simulations on the 1D–1D examples including the smooth and discontinuous initial condition problems, Sod shock tube problem, the shock structure problems, and the 1D–3D examples including the smooth and discontinuous problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.

An Isoperimetric Sloshing Problem in a Shallow Container with Surface Tension

In 1965, B. A. Troesch solved the isoperimetric sloshing problem of determining the container shape that maximizes the fundamental sloshing frequency among two classes of shallow containers: symmetric canals with a given free surface width and cross-sectional area, and radially symmetric containers with a given rim radius and volume (Commun Pure Appl Math 18(1–2):319–338, 1965, https://doi.org/10.1002/cpa.3160180124 ). Here, we extend these results in two ways: (i) we consider surface tension effects on the fluid free surface, assuming a flat equilibrium free surface together with a pinned contact line, and (ii) we consider sinusoidal waves traveling along the canal with wavenumber $$\alpha \ge 0$$ and spatial period $$2\pi /\alpha $$ ; two-dimensional sloshing corresponds to the case $$\alpha = 0$$ . Generalizing our recent variational characterization of fluid sloshing with surface tension to the case of a pinned contact line, we derive the pinned-edge linear shallow sloshing problem, which is an eigenvalue problem for a generalized Sturm-Liouville system. In the case without surface tension, we show that the optimal shallow canal is a rectangular canal for any $$\alpha > 0$$ . In the presence of surface tension, we solve for the maximizing cross-section explicitly for shallow canals with any given $$\alpha \ge 0$$ and shallow radially symmetric containers with m azimuthal nodal lines, $$m = 0, 1$$ . Our results reveal that the squared maximal sloshing frequency increases considerably as surface tension increases. Interestingly, both the optimal shallow canal for $$\alpha = 0$$ and the optimal shallow radially symmetric container are not convex. As a consequence of our explicit solutions, we establish convergence of the maximizing cross-sections, as surface tension vanishes, to the maximizing cross-sections without surface tension.

A Bochner formula on path space for the Ricci flow

We generalize the classical Bochner formula for the heat flow on evolving manifolds $$(M,g_{t})_{t \in [0,T]}$$ to an infinite-dimensional Bochner formula for martingales on parabolic path space $$P{\mathcal {M}}$$ of space-time $${\mathcal {M}} = M \times [0,T]$$ . Our new Bochner formula and the inequalities that follow from it are strong enough to characterize solutions of the Ricci flow. Specifically, we obtain characterizations of the Ricci flow in terms of Bochner inequalities on parabolic path space. We also obtain gradient and Hessian estimates for martingales on parabolic path space, as well as condensed proofs of the prior characterizations of the Ricci flow from Haslhofer–Naber (J Eur Math Soc 20(5):1269–1302, 2018a). Our results are parabolic counterparts of the recent results in the elliptic setting from Haslhofer–Naber (Commun Pure Appl Math 71(6):1074–1108, 2018b).

Macroscopic Behaviour in a Two-Species Exclusion Process Via the Method of Matched Asymptotics

We consider a two-species simple exclusion process on a periodic lattice. We use the method of matched asymptotics to derive evolution equations for the two population densities in the dilute regime, namely a cross-diffusion system of partial differential equations for the two species’ densities. First, our result captures non-trivial interaction terms neglected in the mean-field approach, including a non-diagonal mobility matrix with explicit density dependence. Second, it generalises the rigorous hydrodynamic limit of Quastel (Commun Pure Appl Math 45(6):623–679, 1992), valid for species with equal jump rates and given in terms of a non-explicit self-diffusion coefficient, to the case of unequal rates in the dilute regime. In the equal-rates case, by combining matched asymptotic approximations in the low- and high-density limits, we obtain a cubic polynomial approximation of the self-diffusion coefficient that is numerically accurate for all densities. This cubic approximation agrees extremely well with numerical simulations. It also coincides with the Taylor expansion up to the second-order in the density of the self-diffusion coefficient obtained using a rigorous recursive method.

Complete Classification of Gradient Blow-Up and Recovery of Boundary Condition for the Viscous Hamilton–Jacobi Equation

It is known that the Cauchy–Dirichlet problem for the superquadratic viscous Hamilton–Jacobi equation $$u_t-\Delta u=|\nabla u|^p$$ with $$p>2$$ , which has important applications in stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after the appearance of singularity in finite time, which occurs through gradient blow-up (GBU) on the boundary. The solutions eventually become classical again for large time, but in-between they may undergo losses and recoveries of boundary conditions at multiple times (as well as GBU at multiple times), hence forming a quite complex dynamics. In this paper, with the help of braid group theory combined with inner/outer region analysis, we give a complete classification of all viscosity solutions in one-dimensional case. Namely, we fully describe the (countably many, type II) rates and space–time profiles of gradient blow-up (GBU) or recovery of boundary condition (RBC), we characterize them at any time when such a phenomenon occurs, and we moreover determine whether the space–time profiles are stable or unstable. These results can be modified in radial domains in general dimensions. Even for type II blow-up in other PDEs, as far as we know, there has been no complete classification except Mizoguchi (Commun Pure Appl Math 75:1870–1886, 2022), in which the argument relies on features peculiar to chemotaxis system. Whereas there are many results on construction of special type II blow-up solutions of PDEs with investigation of stability/instability of bubble, determination of stability/instability of space–time profile for general solutions has not been done. A key in our proofs is to focus on braid group algebraic structure with respect to vanishing intersections with the singular steady state, as a GBU or RBC time is approached. In turn, the GBU and RBC rates and profiles, as well as their stability/instability, can be given a complete geometric characterization by the number of vanishing intersections. We construct special solutions in bounded and unbounded intervals in both GBU and RBC cases, based on methods from Herrero and Velázquez (Preprint, 1994), and then we apply braid group theory to get upper and lower estimates of the rates. After that, we rule out oscillation of the rates, which leads us to the complete space–time profile. In the process, careful construction of special solutions with specific behaviors in intermediate and outer regions, which is far from bubble and the RBC point, plays an essential role. The application of such techniques to viscosity solutions is completely new.

Stochastic homogenization: a short proof of the annealed Calderón–Zygmund estimate

A building block for many field theories in continuum physics are second-order elliptic operators in divergence form, as given through a coefficient field which may be assimilated to a metric tensor field on \(\mathbb {R}^d\). The mapping properties of these linear operators are a crucial ingredient for analysis. In this paper, we focus on Calderón–Zygmund estimates, that is, on the boundedness of the corresponding Helmholtz projection in \(\mathrm{{L}}^p(\mathbb {R}^d)\)-spaces. Even when the coefficient field is uniformly smooth, this estimate may fail for p not close to 2. We seek an intrinsic criterion on the validity of the Calderón–Zygmund estimate in the whole range of \(p\in (1,\infty )\); intrinsic in the sense that it is formulated in terms of the scalar and vector potentials of the harmonic coordinates. We seek genericity in form of a statistical statement, and thus consider general ensembles of coefficient fields. Our criterion comes in form of finite stochastic moments for the potentials, or rather their corrections from being affine. In line with this, the Calderón–Zygmund estimates we obtain are annealed as opposed to quenched, meaning that there is an inner norm in form of a stochastic moment next to the (outer) \(\mathrm{{L}}^p\)-norm in space. This result grows out of recent progress in quantitative stochastic homogenization; it is ultimately inspired by the classical large-scale regularity theory of Avellaneda and Lin (Commun Pure Appl Math 40(6):803–847, 1987). More specifically, we provide an easier version of the proof given by us in Josien and Otto (J Funct Anal, 2022), albeit under stronger assumptions. Annealed Calderon–Zygmund estimates were first established in Duerinckx and Otto (Stoch Partial Differ Equ Anal Comput 8(3):625–692, 2020), and are a very convenient tool for error estimates in stochastic homogenization.

Invariance Principles and Log-Distance of F-KPP Fronts in a Random Medium

We study the front of the solution to the F-KPP equation with randomized non-linearity. Under suitable assumptions on the randomness including spatial mixing behavior and boundedness, we show that the front of the solution lags at most logarithmically in time behind the front of the solution of the corresponding linearized equation, i.e. the parabolic Anderson model. This can be interpreted as a partial generalization of Bramson’s findings (Bramson in Commun Pure Appl Math 31(5):531–581, 1978) for the homogeneous setting. Partially building on this result and its derivation, we establish functional central limit theorems for the fronts of the solutions to both equations.

Dimension reduction through gamma convergence for general prestrained thin elastic sheets

We study thin films with residual strain by analyzing the Gamma -limit of non-Euclidean elastic energy functionals as the material’s thickness tends to 0. We begin by extending prior results (Bhattacharya et al. in Arch Ration Mech Anal 228: 143–181, 2016); (Agostiniani et al. in ESAIM Control Opt Calculus Var 25: 24, 2019); (Lewicka and Lucic in Commun Pure Appl Math 73: 1880–1932, 2018); (Schmidt in J de Mathématiques Pures et Appliquées 88: 107–122, 2007) , to a wider class of films, whose prestrain depends on both the midplate and the transversal variables. The ansatz for our Gamma -convergence result uses a specific type of wrinkling, which is built on exotic solutions to the Monge-Ampere equation, constructed via convex integration (Lewicka and Pakzad in Anal PDE 10: 695–727, 2017). We show that the expression for our Gamma -limit has a natural interpretation in terms of the orthogonal projection of the residual strain onto a suitable subspace. We also show that some type of wrinkling phenomenon is necessary to match the lower bound of the Gamma -limit in certain circumstances. These results all assume a prestrain of the same order as the thickness; we also discuss why it is natural to focus on that regime by considering what can happen when the prestrain is larger.

Mass Scaling of the Near-Critical 2D Ising Model Using Random Currents

We examine the Ising model at its critical temperature with an external magnetic field \(h a^{\frac{15}{8}}\) on \(a\mathbb {Z}^2\) for \(a,h >0\). A new proof of exponential decay of the truncated two-point correlation functions is presented. It is proven that the mass (inverse correlation length) is of the order of \(h^\frac{8}{15}\) in the limit \(h \rightarrow 0\). This was previously proven with CLE-methods in Camia et al. in (Commun Pure Appl Math 73(7):1371–405, 2020). Our new proof uses instead the random current representation of the Ising model and its backbone exploration. The method further relies on recent couplings to the random cluster model (Aizenman et al. in Invent Math 216:661–743, 2018) as well as a near-critical RSW-result for the random cluster model (Duminil-Copin and Manolescu in Planar Random-Cluster Model: Scaling Relations, 2020).

Harnack inequality and principal eigentheory for general infinity Laplacian operators with gradient in $$\mathbb {R}^N$$ and applications

Under the lack of variational structure and nondegeneracy, we investigate three notions of generalized principal eigenvalue for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality is proved to support our analysis. This is a continuation of our first work (Biswas and Vo in Liouville theorems for infinity Laplacian with gradient and KPP type equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. https://doi.org/10.2422/2036-2145.202105_050) and a contribution in the development of the theory of generalized principal eigenvalue beside the works (Berestycki et al. in Commun Pure Appl Math 47(1):47–92, 1994; Berestycki and Rossi in JEMS 8:195–215, 2006; Berestycki and Rossi in Commun Pure Appl Math 68(6):1014–1065, 2015; Berestycki et al. in J Math Pures Appl 103:1276–1293, 2015; Nguyen and Vo in Calc Var Partial Differ Equ 58(3):102 2019). We use these notions to characterize the validity of maximum principle and study the existence, nonexistence and uniqueness of positive solutions of Fisher-KPP type equations in the whole space. The sliding method is intrinsically improved for infinity Laplacian to solve the problem. The results are related to the Liouville type results, which will be meticulously explained.

KPZ equation from non-simple variations on open ASEP

This paper has two main goals. The first is universality of the KPZ equation for fluctuations of dynamic interfaces associated to interacting particle systems in the presence of open boundary. We consider generalizations on the open-ASEP from Corwin and Shen (Commun Pure Appl Math 71(10):2065–2128, 2018), Parekh (Commun Math Phys 365:569–649, 2019. https://doi.org/10.1007/s00220-018-3258-x). but admitting non-simple interactions both at the boundary and within the bulk of the particle system. These variations on open-ASEP are not integrable models, similar to the long-variations on ASEP considered in Dembo and Tsai (Commun Math Phys 341(1):219–261, 2016), Yang (Kardar–Parisi–Zhang equation from long-range exclusion processes, 2020. arXiv:2002.05176 [math.PR]). We establish the KPZ equation with the appropriate Robin boundary conditions as scaling limits for height function fluctuations associated to these non-integrable models, providing further evidence for the aforementioned universality of the KPZ equation. We specialize to compact domains and address non-compact domains in a second paper (Yang in KPZ equation from non-simple dynamics with boundary in the non-compact regime). The procedure that we employ to establish the aforementioned theorem is the second main point of this paper. Invariant measures in the presence of boundary interactions generally lack reasonable descriptions. Thus, global analyses done through the invariant measure, including the theory of energy solutions in Goncalves and Jara (Arch Ration Mech Anal 212:597, 2014), Goncalves and Jara (Stoch Process Appl 127(12):4029–4052, 2017), Goncalves et al. (Ann Probab 43(1):286–338, 2015), is immediately obstructed. To circumvent this obstruction, we appeal to the almost entirely local nature of the analysis in Yang (2020).

Local Well-Posedness and Incompressible Limit of the Free-Boundary Problem in Compressible Elastodynamics

We consider the three dimensional free-boundary compressible elastodynamic system under the Rayleigh–Taylor sign condition. This describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor is assumed to satisfy the neo-Hookean linear elasticity. The local well-posedness was proved by Trakhinin (J Differ Eq 264(3):1661–1715, 2018) by Nash–Moser iteration. In this paper, we give a new proof of the local well-posedness by the combination of classical energy method and hyperbolic approach. In the proof, we apply the tangential smoothing method to define the approximation system. The key observation is that the structure of the wave equation of pressure together with Christodoulou–Lindblad (Commun Pure Appl Math 53(12):1536–1602, 2000) elliptic estimates reduces the energy estimates to the control of tangentially-differentiated wave equations despite a potential loss of derivative in the source term. To the best of our knowledge, we first establish the nonlinear energy estimate without loss of regularity for free-boundary compressible elastodynamics. The energy estimate is also uniform in sound speed which yields the incompressible limit, that is, the solutions of the free-boundary compressible elastodynamic equations converge to the incompressible counterpart provided the convergence of initial datum. It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to include the full time derivatives in the boundary energy and analyze higher order wave equations as in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) can still be recovered for a slightly compressible elastic medium by further delicate analysis of the Alinhac good unknowns, which is completely different from Euler equations.