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In this paper, we study the Cauchy problem for systems of 3-D quasilinear wave equations satisfying the null condition with initial data of low regularity. In the radially symmetric case, we prove the global existence for every small data in H 3 × H 2 with a low weight. To achieve this goal, we will show how to extend the global iteration method first suggested by Li and Chen (1988) [32] to the low regularity case, which is also another purpose of this paper. Finally, we apply our result to 3-D nonlinear elastic waves.

International audience; In this paper, we consider a semi-linear fourth order parabolic equation in a bounded smooth domain Ω with homogeneous Dirichlet and Neumann boundary conditions. The main result of this paper is the null controllability and the exact controllability to the trajectories at any time T > 0 for the associated control system with a control function acting at the interior.; Dans ce papier, on considère uneéquation parabolique semi-linéaire de quatrième ordre dans un domaine borné régulier Ω avec des conditions aux limites de type Dirichlet et Neumann homogènes. Le résultat principal de ce papier concerne la contrôlabilitéà zéro et la contrôlabilité exacte pour tout T > 0 du système de contrôle associé avec un contrôle agissantà l'interieur.

We investigate a steady planar flow of an ideal fluid in a bounded simply connected domain and focus on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence of solutions for this problem: the vorticity method and the stream function method. A long standing open problem is whether these two entirely different methods result in the same solution. In this paper, we will give a positive answer to this problem by studying the local uniqueness of the solutions. Another result obtained in this paper is that if the domain is convex, then the vortex patch problem has a unique solution.

The purpose of this paper is to show well-posedness results in L 2 -based Sobolev spaces for transmission, Dirichlet, Neumann, and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on a compact Riemannian manifold of dimension m ≥ 2 . The Dirichlet, transmission, and mixed problems for the nonlinear Darcy-Forchheimer-Brinkman system with L ∞ coefficients are also analyzed. First, we focus on the well-posedness of linear transmission, Dirichlet and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using a variational approach that reduces such a boundary value problem to a mixed variational formulation defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. Further, we show the equivalence between each boundary value problem for the Brinkman system with L ∞ coefficients and its mixed variational counterpart, and then the well posedness in L 2 -based Sobolev spaces by using the Necas-Babuska-Brezzi technique. The second goal of this paper is the construction of the Newtonian and layer potential operators for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using the well-posedness results for the analyzed linear transmission problems. Various mapping properties of these operators are also obtained and used to describe the weak solutions of the Poisson problems with Dirichlet and Neumann conditions for the nonsmooth Brinkman system in terms of such potentials. Finally, we combine the well-posedness results of the Poisson problems of Dirichlet, transmission, and mixed type for the nonsmooth Brinkman system with a fixed point theorem in order to show the existence of a weak solution of the Poisson problem of Dirichlet, transmission, or mixed type for the (nonlinear) Darcy-Forchheimer-Brinkman system with L ∞ coefficients in L 2 -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds of dimension m ∈ { 2 , 3 } .

This paper is concerned with the asymptotic behavior of bounded solutions of the Cauchy problem { u t = u x x + f ( t , u ) , x ∈ R , t > 0 , u ( x , 0 ) = u 0 , x ∈ R , where u 0 is a nonnegative bounded function with compact support and f is a rather general nonlinearity that is periodic in t and satisfies f ( ⋅ , 0 ) = 0 . In the autonomous case where f = f ( u ) , the convergence of every bounded solution to an equilibrium has been established by Du and Matano (2010). However, the presence of periodic forcing makes the problem significantly more difficult, partly because the structure of time periodic solutions is much less understood than that of steady states. In this paper, we first prove that any ω-limit solution is either spatially constant or symmetrically decreasing. Furthermore, we show that the set of ω-limit solutions either consists of a single time-periodic solution or it consists of multiple time-periodic solutions and heteroclinic connections among them. Next, under a mild non-degenerate assumption on the corresponding ODE, we prove that the ω-limit set is a singleton, which implies the solution converges to a time-periodic solution. Lastly, we apply these results to equations with bistable nonlinearity and combustion nonlinearity, and specify more precisely which time-periodic solutions can possibly be selected as the limit.

Though the boundary layer formation in the chemotactic process has been observed in experiment (cf. [63] ), the mathematical study on the boundary layer solutions of chemotaxis models is just in its infant stage. Apart from the sophisticated theoretical tools involved in the analysis, how to impose/derive physical boundary conditions is a state-of-the-art in studying the boundary layer problem of chemotaxis models. This paper will proceed with a previous work [24] in one dimension to establish the convergence of boundary layer solutions of the Keller–Segel model with singular sensitivity in a two-dimensional space (half-plane) with respect to the chemical diffusion rate denoted by e ≥ 0 . Compared to the one-dimensional boundary layer problem, there are many new issues arising from multi-dimensions such as possible Prandtl type degeneracy, curl-free preservation and well-posedness of large-data solutions. In this paper, we shall derive appropriate physical boundary conditions and gradually overcome these barriers and hence establish the convergence of boundary layer solutions of the singular Keller–Segel system in the half-plane as the chemical diffusion rate vanishes. Specially speaking, we justify that the boundary layer converges to the outer layer (solution with e = 0 ) plus the inner layer as e → 0 , where both outer and inner layer profiles are precisely derived and well understood. By doing this, the structure of boundary layer solutions is clearly characterized. We hope that our results and methods can shed lights on the understanding of underlying mechanisms of the boundary layer patterns observed in the experiment for chemotaxis such as the work by Tuval et al. [63] , and open a new window in the future theoretical study of chemotaxis models.

This paper deals with the existence of multiple solutions for the quasilinear equation − div A ( x , ∇ u ) + V ( x ) | u | α ( x ) − 2 u = f ( x , u ) in R N , which involves a general variable exponent elliptic operator in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like | ξ | q ( x ) − 2 ξ for small | ξ | and like | ξ | p ( x ) − 2 ξ for large | ξ | , where 1 α ( ⋅ ) ≤ p ( ⋅ ) q ( ⋅ ) N . Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works A. Azzollini et al. (2014) [4] and N. Chorfi and V. Radulescu (2016) [11] from cases where the exponents p and q are constant, to the case where p ( ⋅ ) and q ( ⋅ ) are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the Cerami compactness condition.

Motivated by the paper Gerard-Varet and Dormy (2010) [6] [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equations in Sobolev spaces, this paper aims to extend the result in [6] to the case when the shear flow has general decay. The key observation is to construct an approximate solution that captures the initial layer to the linearized problem motivated by the precise formulation of solutions to the inviscid Prandtl equations.

This paper concerns approximate cloaking by mapping for a full, but scalar wave equation, when one allows for physically relevant frequency dependence of the material properties of the cloak. The paper is a natural continuation of [20] , but here we employ the Drude–Lorentz model in the cloaking layer, that is otherwise constructed by an approximate blow up transformation of the type introduced in [10] . The central mathematical problem is to analyze the effect of a small inhomogeneity in the context of this non-local full wave equation.

We consider nonlinear diffusion problems of the form ut=Δu+f(u) with Stefan type free boundary conditions, where the nonlinear term f(u) is of monostable, bistable or combustion type. Such problems are used as an alternative model (to the corresponding Cauchy problem) to describe the spreading of a biological or chemical species, where the free boundary represents the expanding front. We are interested in its long-time spreading behavior which, by recent results of Du, Matano and Wang [10], is largely determined by radially symmetric solutions. Therefore we will examine the radially symmetric case, where the equation is satisfied in |x| 0. Subsequently, sharper estimate of the spreading speed was obtained by the authors of the current paper in [11], in the form that limt→∞⁡[h(t)−c⁎t]=Hˆ∈R1. In this paper, we consider the case N≥2 and show that a logarithmic shifting occurs, namely there exists c⁎>0 independent of N such that limt→∞⁡[h(t)−c⁎t+(N−1)c⁎log⁡t]=hˆ∈R1. At the same time, we also obtain a rather clear description of the spreading profile of u(t,r). These results reveal striking differences from the spreading behavior modeled by the corresponding Cauchy problem.

We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a very rough thin layer and embedded in an ambient medium. The roughness of the layer is described by a quasi ε-periodic function, ε being a small parameter, while the mean thickness of the layer is of magnitude εβ, where β∈(0,1). Using the two-scale analysis, we replace the very rough thin layer by appropriate transmission conditions on the boundary of the object, which lead to an explicit characterization of the polarization tensor as defined in Vogelius and Capdeboscq [Y. Capdeboscq, M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN Math. Model. Numer. Anal. 37 (2003) 159–173]. The main result of this paper is quite unexpected, and the approximate transmission conditions are not intuitive since they mix in a complex way both conductivities of the exterior medium and of the membrane. This paper extends the previous works of Poignard [C. Poignard, Approximate transmission conditions through a weakly oscillating thin layer, Math. Meth. Appl. Sci. 32 (2009) 435–453] and of Ciuperca et al. [I. Ciuperca, M. Jai, C. Poignard, Approximate transmission conditions through a rough thin layer. The case of periodic roughness, Euro. J. Appl. Math. 21 (2010) 51–75], in which β⩾1.

The starting point of this work is a paper by Alvarez, Lasry and Lions (1997) concerning the convexity and the partial convexity of solutions of fully nonlinear degenerate elliptic equations. We extend their results in two directions. First, we deal with possibly sublinear (but epi-pointed) solutions instead of 1-coercive ones; secondly, the partial convexity of C 2 solutions is extended to the class of continuous viscosity solutions. A third contribution of this paper concerns C 1 , 1 estimates for convex viscosity solutions of strictly elliptic nonlinear equations. To finish with, all the tools and techniques introduced here permit us to give a new proof of the Alexandroff estimate obtained by Trudinger (1988) and Caffarelli (1989).

The main result of this paper provides uniform decay rates obtained for the energy function associated with a three-dimensional structural acoustic model described by coupled system consisting of the wave equation and plate equation with the coupling on the interface between the acoustic chamber and the wall. The uniform stabilization is achieved by introducing a nonlinear dissipation acting via boundary forces applied at the edge of the plate and viscous or boundary damping applied to the wave equation. The results obtained in this paper extend, to the non-analytic, hyperbolic-like setting, the results obtained previously in the literature for acoustic problems modeled by structurally damped plates (governed by analytic semigroups). As a bypass product, we also obtain optimal uniform decay rates for the Euler Bernoulli plate equations with nonlinear boundary dissipation acting via shear forces only and without (i) any geometric conditions imposed on the domain ,(ii) any growth conditions at the origin imposed on the nonlinear function. This is in contrast with the results obtained previously in the literature ([22] and references therein).

This paper is dedicated to the effective viscosity of suspensions without inertia, at low solid volume fraction ϕ. The goal is to derive rigorously a o ( ϕ 2 ) formula for the effective viscosity. In [17] , [19] , such formula was given for rigid spheres satisfying the strong separation assumption d m i n ≥ c ϕ − 1 3 r , where d m i n is the minimal distance between the spheres and r their radius. It was then applied to both periodic and random configurations with separation, to yield explicit values for the O ( ϕ 2 ) coefficient. We consider here complementary (and certainly more realistic) random configurations, satisfying softer assumptions of separation, and long range decorrelation. We justify in this setting the famous Batchelor-Green formula [3] . Our result applies for instance to hardcore Poisson point process with almost minimal hardcore assumption d m i n > ( 2 + e ) r , e > 0 .

In this paper, we consider the exact boundary controllability of nodal profile for Saint-Venant systems on a network with loops. In the case with one object-node, the proposed cut-off method not only contains the results obtained in [13] on a special network with a triangle-like loop in a unified way, but also generalizes these results to general networks with loops. Based on the results in the case with one object-node, the proposed cut-off method can also be applied successfully to the case with several object-nodes.

This paper studies a dissipative heat equation with Delta sources of non-linear strength located on a periodic lattice. The model arises from intracellular waves in continuum excitable media with discrete release sites. Due to the presence of Delta sources, the solution of the model has discontinuous spatial derivatives. We focus on the bistable regime of the model, determined by the decay strength parameter a and the separation distance L between release sites, in which the model admits exactly three L-periodic steady states. We establish the existence of pulsating waves spatially connecting them. For the case of waves connecting two stable L-periodic steady states, the uniqueness and global exponential stability of pulsating waves are shown. Also a new technique is introduced to find the fine structure of the tails of pulsating waves.

We use geometric microlocal methods to compute an asymptotic expansion of mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. This paper can be seen as the Riemannian 3-manifold version of the planar result of [1] and thus enable us to see the full effect of the local extrinsic boundary geometry on the mean arrival time of the Brownian particles. Our approach also connects this question to some of the recent progress on boundary rigidity and integral geometry [21] and [18] .

In composite materials, the stress may be arbitrarily large in the narrow region between two close-to-touching hard inclusions. The stress is represented by the gradient of a solution to the Lame system of linear elasticity. The aim of this paper is to establish lower bounds of the gradients of solutions of the Lame system with partially infinite coefficients as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero. Combining it with the pointwise upper bounds obtained in our previous work, the optimality of the blow-up rate of gradients is proved for inclusions with arbitrary shape in dimensions two and three. The key to show this is that we find a blow-up factor, a linear functional of the boundary data, to determine whether the blow-up will occur or not.

We consider the initial boundary value problem for the isentropic compressible Navier-Stokes equations with cylindrical symmetry. The existence of boundary layers is well-known when the shear viscosity vanishes. In this paper, we derive explicit Prandtl type boundary layer equations and prove the global in time stability of the boundary layer profile together with the optimal convergence rate of the vanishing shear viscosity limit without any smallness assumption on the initial and boundary data.

We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational work (Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 63-113), symmetric quasiconvexity is the foremost notion as to sequential weak*-lower semicontinuity of functionals on BD. The overarching main difficulty here is Ornstein's Non-Inequality, implying that the BD-case is genuinely different from the study of variational integrals on BV. In particular, this paper extends the recent work of Kristensen and the author (Partial regularity for BV-Minimizers, Arch. Ration. Mech. Anal. 232 (2019), Issue 3, 1429-1473) from the BV- to the BD-situation. Alongside, we establish partial regularity results for strongly quasiconvex functionals of superlinear growth by reduction to the full gradient case, which might be of independent interest., Comment: Version 2, 40 pages, 1 figure, final version to appear at J. Math. Pures Appl

This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary Holder regularity under proper geometric conditions. “Unified” means that our method is applicable for the Laplace equation, linear elliptic equations in divergence and non-divergence form, fully nonlinear elliptic equations, the p−Laplace equations and the fractional Laplace equations etc. In addition, these geometric conditions are quite general. In particular, for local equations, the measure of the complement of the domain near the boundary point concerned could be zero. The key observation in the method is that the strong maximum principle implies a decay for the solution, then a scaling argument leads to the Holder regularity. Moreover, we also give a geometric condition, which guarantees the solvability of the Dirichlet problem for the Laplace equation. The geometric meaning of this condition is more apparent than that of the Wiener criterion.

In this paper we study the structure of positive vector solutions for nonlinear Schrodinger systems with 3 components when all interspecies interaction forces are positive and large while all intraspecies interaction forces are positive and fixed. We will show that the structure strongly depends on some relation of large interspecies interaction forces.

The purpose of this paper is three-fold: the first is to determine the Campanato-Sobolev space I N ( L p , κ ) by means of ∑ | α | = N ‖ D α f ‖ L p , κ - the sum of the Campanato norms of the derivatives D α f with | α | = N ; the second is to characterize Morrey's fractional integrals { T f : f ∈ L p , κ } in the Campanato-Sobolev space I s ( L p , κ ) ; the third is to find a distributional solution F ∈ ( L q , λ ) n of the mean curvature type divergence equation div F = f ∈ L p , κ (the Morrey space). And yet, the here-established three theorems and their proofs are not only novel but also nontrivial.

We consider the following k-coupled nonlinear Schrodinger systems: { − Δ u j + λ j u j = μ j u j 3 + ∑ i = 1 , i ≠ j k β i , j u i 2 u j in R N , u j > 0 in R N , u j ( x ) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , k , where N ≤ 3 , k ≥ 3 , λ j , μ j > 0 are constants and β i , j = β j , i ≠ 0 are parameters. There have been intensive studies for the above systems when k = 2 or the systems are purely attractive ( β i , j > 0 , ∀ i ≠ j ) or purely repulsive ( β i , j 0 , ∀ i ≠ j ); however very few results are available for k ≥ 3 when the systems admit mixed couplings and the components are organized into groups, i.e., there exist ( i 1 , j 1 ) and ( i 2 , j 2 ) such that β i 1 , j 1 > 0 and β i 2 , j 2 0 . In this paper we give the first systematic and an (almost) complete study on the existence of ground states when the systems admit mixed couplings and the components are organized into groups. We first divide these systems into repulsive-mixed and total-mixed cases. In the first case we prove nonexistence of ground states. In the second case we give a necessary condition for the existence of ground states and also provide estimates for Morse index. The key idea is the block decomposition of the systems (optimal block decompositions, eventual block decompositions), and the measure of total interaction forces between different blocks. Finally the assumptions on the existence of ground states are shown to be optimal in some special cases.

This paper is concerned with the Minkowski convolution of viscosity solutions of fully nonlinear parabolic equations. We adopt this convolution to compare viscosity solutions of initial-boundary value problems in different domains. As a consequence, we can for instance obtain parabolic power concavity of solutions to a general class of parabolic equations. Our results are applicable to the Pucci operator, the normalized q-Laplacians with 1 q ≤ ∞ , the Finsler Laplacian, and more general quasilinear operators.

In this paper, we consider the Neumann–Poincare type operator associated with the Lame system of linear elasticity. It is known that if the boundary of a planar domain is smooth enough, it has eigenvalues converging to two different points determined by Lame parameters. We show that eigenvalues converge at a polynomial rate on smooth boundaries and the convergence rate is determined by smoothness of the boundary. We also show that they converge at an exponential rate if the boundary of the domain is real analytic.

In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed., Comment: 25 papges, 2 figures

In this paper, we establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through introducing a new method of cutting off the stochastic system and using a fixed point argument in a carefully constructed Banach space. To get the weak solution we first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution.

We propose a general model to investigate the joint impact of viral diffusion and cell-to-cell transmission on viral dynamics. The mathematical challenge lies in the fact that the model system is partially degenerate and the solution map is not compact. While the simpler cases with only indirect transmission mode or weak cell-to-cell transmission mode have been extensively studied in the literature, it remains an open problem to understand the local and global dynamics of fully coupled viral infection model with partial degeneracy. In this paper, we identify the basic reproduction number as the spectral radius of the sum of two linear operators corresponding to direct and indirect transmission modes. It is well-known that viral mobility may induce infection in low-risk regions. However, as diffusion coefficient increases, we prove that the basic reproduction number actually decreases, which indicates that faster viral movements may result in a lower level of viral infection. By an innovative construction of Lyapunov functionals, we further demonstrate that the basic reproduction number is the threshold parameter which determines global picture of viral dynamics. In addition to the traditional dichonomy results of extinction and persistence as obtained in earlier works for many simpler models, we are able to prove global asymptotic stability of infection-free steady state and global attractiveness (as well as uniqueness) of chronic-infection steady state, depending on whether the basic reproduction number is smaller or greater than one. Numerical simulation supports our theoretical results and suggests an interesting phenomenon: boundary layer and internal layer may occur when the diffusion parameter tends to zero.

This paper is concerned with blowup in a parabolic-parabolic system describing chemotactic aggregation. In a disk, radial solutions blow up in finite time if their initial energy is less than some value. In the whole plane, the energy diverges to −∞ as time goes to +∞ for any forward selfsimilar solution. This implies that one cannot expect to get a sufficient condition for finite-time blowup using energy as in a disk. For a solution ( u , v ) , u and v denote density of cells and of chemical substance, respectively. Let τ be the coefficient of time derivative of v. We first prove that for τ > 0 there exists M ( τ ) > 0 with M ( τ ) → ∞ as τ → ∞ such that all radial solutions ( u , v ) with initial mass of u larger than M ( τ ) blow up in finite time. On the other hand, it was shown in [22] that any blowup in the system with τ = 1 is type II (not necessarily in radial case). Removing the restriction on τ, we get the conclusion for all τ > 0 .

In this paper, the global well-posedness of semirelativistic equations with a power type nonlinearity on Euclidean spaces is studied. In two dimensional H s scaling subcritical case with 1 ≤ s ≤ 2 , the local well-posedness follows from a Strichartz estimate. In higher dimensional H 1 scaling subcritical case, the local well-posedness for radial solutions follows from a weighted Strichartz estimate. Moreover, in three dimensional H 1 scaling critical case, the local well-posedness for radial solutions follows from a uniform bound of solutions which may be derived by the corresponding one dimensional problem. Local solutions may be extended by a priori estimates.

This paper studies sign-changing solutions and their concentration behaviors of logarithmic scalar field equations in the semiclassical setting. At a local minimum of the potential function we construct an unbounded sequence of sign-changing solutions concentrating near the local minimum. This resembles a localized phenomenon of an unbounded sequence of localized bound states in the studies of nonlinear eigenvalues and eigenfunctions.

The geometric properties of (almost) minimal sets, especially the regularity, is an interesting topic in geometric measure theory, which were often studied in the literature, for example [2] , [4] , [5] , [8] , [9] , [10] , [19] , [22] . Soap films as well as solutions to Plateau's Problem could be a typical example of such kind of minimal sets. In this paper, we will investigate the convergence of a sequence of such sets, and show that Hausdorff convergence and varifold convergence coincide on the class of almost minimal sets bounded by a uniform gauge function, and we will see that a large amount of sets, including all compact C 1 , 1 submanifolds in R n , are almost minimal.

In this paper, we study different notions of solutions of nonlocal and nonlinear equations of fractional $p$-Laplace type $${\rm P.V.} \int_{\mathbb R^n}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dy = 0.$$ Solutions are defined via integration by parts with test functions, as viscosity solutions or via comparison. Our main result states that for bounded solutions, the three different notions coincide., Comment: 21 pages, to appear in Journal de Math\'ematiques Pures et Appliqu\'ees

The exact boundary controllability for hyperbolic systems can not be realized generally on a network with loops (see [16] ). In this paper we consider the exact boundary controllability of nodal profile on a network with loops. Precisely speaking, on a network with a triangle-like loop, when nodal profiles are given at various kinds of nodes, different constructive methods can be used to get the corresponding exact boundary controllability of nodal profile for Saint-Venant system by means of boundary controls acting on suitable nodes, respectively. This reveals that the exact boundary controllability of nodal profile is quite different from the usual exact boundary controllability, and has relatively distinctive behaviors and characters.

In this paper, we prove that the spectral curve Γ n of the generalized Lame equation with the Treibich–Verdier potential y ″ ( z ) = [ ∑ k = 0 3 n k ( n k + 1 ) ℘ ( z + ω k 2 | τ ) + B ] y ( z ) , n k ∈ Z ≥ 0 can be embedded into the symmetric space Sym N E τ of the N-th copy of the torus E τ , where N = ∑ n k . This embedding induces an addition map σ n ( ⋅ | τ ) from Γ n onto E τ . The main result is to prove that the degree of σ n ( ⋅ | τ ) is equal to ∑ k = 0 3 n k ( n k + 1 ) / 2 . This is the first step toward constructing the pre-modular form associated with this generalized Lame equation.

The Boussinesq a b c d system is a 4-parameter set of equations posed in R t × R x , originally derived by Bona, Chen and Saut [11] , [12] as first order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation [17] . Among many particular regimes, depending each of them in terms of the value of the parameters ( a , b , c , d ) present in the equations, the generic regime is characterized by the setting b , d > 0 and a , c 0 . If additionally b = d , the a b c d system is Hamiltonian. The equations in this regime are globally well-posed in the energy space H 1 × H 1 , provided one works with small solutions [12] . In this paper, we investigate decay and the scattering problem in this regime, which is characterized as having (quadratic) long-range nonlinearities, very weak linear decay O ( t − 1 / 3 ) because of the one dimensional setting, and existence of non scattering solutions (solitary waves). We prove, among other results, that for a sufficiently dispersive a b c d systems (characterized only in terms of parameters a , b and c), all small solutions must decay to zero, locally strongly in the energy space, in proper subset of the light cone | x | ≤ | t | . We prove this result by constructing three suitable virial functionals in the spirit of works [27] , [28] , and more precisely [42] (valid for the simpler scalar “good Boussinesq” model), leading to global in time decay and control of all local H 1 × H 1 terms. No parity nor extra decay assumptions are needed to prove decay, only small solutions in the energy space.

This paper studies connections among observable sets, the observability inequality, the Holder-type interpolation inequality and the spectral inequality for the heat equation in R n . We present the characteristic of observable sets for the heat equation. In more detail, we show that a measurable set in R n satisfies the observability inequality if and only if it is γ-thick at scale L for some γ > 0 and L > 0 . We also build up the equivalence among the above-mentioned three inequalities. More precisely, we obtain that if a measurable set in R n satisfies one of these inequalities, then it satisfies others. Finally, we get some weak observability inequalities and weak interpolation inequalities where observations are made over a ball.

In this paper we elucidate the connection between various notions of differentiability in the Wasserstein space: some have been introduced intrinsically (in the Wasserstein space, by using typical objects from the theory of Optimal Transport) and used by various authors to study gradient flows, Hamiltonian flows, and Hamilton–Jacobi equations in this context. Another notion is extrinsic and arises from the identification of the Wasserstein space with the Hilbert space of square-integrable random variables on a non-atomic probability space. As a consequence, the classical theory of well-posedness for viscosity solutions for Hamilton–Jacobi equations in infinite-dimensional Hilbert spaces is brought to bear on well-posedness for Hamilton–Jacobi equations in the Wasserstein space.

An important feature of the planetary oceanic dynamics is that the aspect ratio (the ratio of the depth to horizontal width) is very small. As a result, the hydrostatic approximation (balance), derived by performing the formal small aspect ratio limit to the Navier–Stokes equations, is considered as a fundamental component in the primitive equations of the large-scale ocean. In this paper, we justify rigorously the small aspect ratio limit of the Navier–Stokes equations to the primitive equations. Specifically, we prove that the Navier–Stokes equations, after being scaled appropriately by the small aspect ratio parameter of the physical domain, converge strongly to the primitive equations, globally and uniformly in time, and the convergence rate is of the same order as the aspect ratio parameter. This result validates the hydrostatic approximation for the large-scale oceanic dynamics. Notably, only the weak convergence of this small aspect ratio limit was rigorously justified before.

The fundamental theorem of surface theory asserts that a surface in the three-dimensional Euclidean space E 3 can be reconstructed from the knowledge of its two fundamental forms under the assumptions that their components are smooth enough—classically in the space C 2 ( ω ) for the first one and in the space C 1 ( ω ) for the second one—and satisfy the Gauss and Codazzi–Mainardi equations over a simply-connected open subset ω of R 2 ; the surface is then uniquely determined up to proper isometries of E 3 . Then S. Mardare showed in 2005 that this result still holds under the much weaker assumptions that the components of the first form are only in the space W loc 1 , p ( ω ) and those of the second form only in the space L loc p ( ω ) , the components of the immersion defining the reconstructed surface being then in the space W loc 2 , p ( ω ) , p > 2 . The purpose of this paper is to complement this last result as follows. First, under the additional assumption that ω is bounded and has a Lipschitz-continuous boundary, we show that a similar existence and uniqueness theorem holds with the spaces W m , p ( ω ) instead of W loc m , p ( ω ) . Second, we establish a nonlinear Korn inequality on a surface asserting that the distance in the W 2 , p ( ω ) -norm, p > 2 , between two given surfaces is bounded, at least locally, by the distance in the W 1 , p ( ω ) -norm between their first fundamental forms and the distance in the L p ( ω ) -norm between their second fundamental forms. Third, we show that the mapping that uniquely defines in this fashion a surface up to proper isometries of E 3 in terms of its two fundamental forms is locally Lipschitz-continuous.

This paper is concerned with the lifespan estimate of classical solutions with small initial data to the Cauchy problem of semilinear damped wave equations with the Fujita critical exponent. We establish the following sharp upper bound of the lifespan T ( e ) ≤ exp ⁡ ( C e − 2 n ) in higher dimensions ( n ≥ 4 ) , by using the heat kernel as the test function. Then, together with the previous results, a complete result on the sharp lower and upper bound estimates is obtained in this case.

In this paper we study the asymptotic behaviour of an exact controllability problem for a second order linear evolution equation defined in a two-component composite with e-periodic disconnected inclusions of size e. On the interface we prescribe a jump of the solution that varies according to a real parameter γ. In particular, we suppose that − 1 γ ≤ 1 . The case γ = 1 is the most interesting and delicate one, since the homogenized problem is represented by a coupled system of a P.D.E. and an O.D.E., giving rise to a memory effect. Our approach to exact controllability consists in applying the Hilbert Uniqueness Method, introduced by J.-L. Lions, which leads us to the construction of the exact control as the solution of a transposed problem. Our main result proves that the exact control and the corresponding solution of the e-problem converge to the exact control of the homogenized problem and to the corresponding solution respectively.

This paper is concerned with the theoretical study of polariton resonances for linear elasticity governed by the Lame system in R 3 , and their application for cloaking due to anomalous localized resonances. We derive a very general and novel class of elastic structures that can induce polariton resonances. It is shown that if either one of the two convexity conditions on the Lame parameters is broken, then we can construct certain polariton structures that induce resonances. This significantly extends the relevant existing studies in the literature where the violation of both convexity conditions is required. Indeed, the existing polariton structures are a particular case of the general structures constructed in our study. Furthermore, we consider the polariton resonances within the finite frequency regime, and rigorously verify the quasi-static approximation for diametrically small polariton inclusions. Finally, as an application of the newly found structures, we construct a polariton device of the core-shell-matrix form that can induce cloaking due to anomalous localized resonance in the quasi-static regime, which also includes the existing study as a special case.

In this paper, we draw on the ideas of [5] to extend the standard Serrin criterion [17] to an anisotropic version thereof. Because we work on weak solutions instead of strong ones, the functions involved have low regularity. Our method summarizes in a joint use of a uniqueness lemma in low regularity and the existence of stronger solutions. The uniqueness part uses duality in a way quite similar to the DiPerna-Lions theory, first developed in [7]. The existence part relies on L p energy estimates, whose proof may be found in [5], along with an approximation procedure.

In this paper, we prove the uniqueness and generic regularity of the energy conservative solution for a system of wave equations modeling nematic liquid crystal.

This paper examines the global regularity problem on the two-dimensional (2D) incompressible Boussinesq equations with fractional horizontal dissipation and thermal diffusion. The goal is to establish the global existence and regularity for the Boussinesq equations with minimal dissipation and thermal diffusion. By working with this general 1D fractional Laplacian dissipation, we are no longer constrained to the standard partial dissipation and this study will help understand the issue on how much dissipation is necessary for the global regularity. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques such as integration by parts no longer apply and new tools including several anisotropic embedding and interpolation inequalities involving fractional derivatives are derived. These tools allow us to obtain very sharp upper bounds for the nonlinearities.

In this paper, we study the expansion problem which arises as two-dimensional (2D) pseudo-steady supersonic flow turns a sharp corner and expands into vacuum. The problem catches interaction of a centered simple wave and a backward planar rarefaction wave, which is deduced a Goursat problem for 2D self-similar Euler equations for compressible flow. By the methods of characteristic decomposition and invariant regions, we get the hyperbolicity in the wave interaction domain and prior C 1 estimates of solutions to the Goursat problem. The global solution up to the interface of gas with vacuum to the expansion problem is obtained constructively.

In this paper we deal with the local null controllability of the Navier–Stokes system with nonlinear Navier-slip boundary conditions and internal controls having one vanishing component.

In this paper, we consider nonnegative solutions of spatially heterogeneous reaction–diffusion equations in the whole space. Under some assumptions on the initial conditions, including in particular the case of compactly supported initial conditions, we show that, above any arbitrary positive value, the solution is increasing in time at large times. Furthermore, in the one-dimensional case, we prove that, if the equation is homogeneous outside a bounded interval and the reaction is linear around the zero state, then the solution is time-increasing in the whole line at large times. The question of the monotonicity in time is motivated by a medical imagery issue.