Showing papers in "Siam Journal on Mathematical Analysis in 2017"
TL;DR: In the special case of the optimal transport problem, this technique dates back to the early 1970s as mentioned in this paper and was used to approximate solutions of linear programs in the early 1990s.
Abstract: Replacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the...
143 citations
TL;DR: The regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator, and a pointwise integro-differential formula for $(\partial_t-\Delta)^su(t,x)$ and parabolic maximum principles are developed.
Abstract: We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $(\partial_t-\Delta)^su(t,x)=f(t,x)~{for}~0
77 citations
TL;DR: The Cauchy problem for the full compressible Navier--Stokes equations with vanishing of density at infinity in $\mathbb{R}^3$ is considered to prove the existence (and uniqueness) of global strong and classical solutions and study the large-time behavior of the solutions as well as the decay rates in time.
Abstract: We consider the Cauchy problem for the full compressible Navier--Stokes equations with vanishing of density at infinity in $\mathbb{R}^3$. Our main purpose is to prove the existence (and uniqueness) of global strong and classical solutions and study the large-time behavior of the solutions as well as the decay rates in time. Our main results show that the strong solution exists globally in time if the initial mass is small for the fixed coefficients of viscosity and heat conduction, and can be large for the large coefficients of viscosity and heat conduction. Moreover, large-time behavior and a surprisingly exponential decay rate of the strong solution are obtained. Finally, we show that the global strong solution can become classical if the initial data are more regular. Note that the assumptions on the initial density do not exclude that the initial density may vanish in a subset of $\mathbb{R}^3$ and that it can be of a nontrivially compact support. To our knowledge, this paper contains the first resul...
69 citations
TL;DR: An effective medium theory for acoustic wave propagation in a bubbly fluid near the Minnaert resonant frequency is derived and it is shown that the obtained effective media may be highly refractive, which can be used to explain the superfocusing experiment observed in [M. Lanoy et al., Phys. Rev. B, 91 (2015), 224202].
Abstract: We derive an effective medium theory for acoustic wave propagation in a bubbly fluid near the Minnaert resonant frequency. We start with a multiple scattering formulation of the scattering problem in which an incident wave impinges on a large number of identical and small bubbles in a homogeneous fluid. Under certain conditions on the configuration of the bubbles distribution, we justify the point interaction approximation and establish an effective medium theory for the bubbly fluid as the number of bubbles tends to infinity. The convergence rate is also derived. As a consequence, we show that near and below the Minnaert resonant frequency, the obtained effective media may be highly refractive, which can be used to explain the superfocusing experiment observed in [M. Lanoy et al., Phys. Rev. B, 91 (2015), 224202]. Moreover, above the Minnaert resonant frequency, the effective medium is dissipative, while at that frequency, effective medium theory does not hold. Our theory sheds light on the mechanism of ...
60 citations
TL;DR: A population structured by a spacevariable and a phenotypical trait, submitted to dispersion,mutations, growth and nonlocal competition, is considered, which introduces the climate shift due to Global Warming and discusses the dynamic of the population by studying the long time behavior of the Cauchy problem.
Abstract: We consider a population structured by a space
variable and a phenotypical trait, submitted to dispersion,
mutations, growth and nonlocal competition. We introduce the
climate shift due to {\it Global Warming} and discuss the dynamics
of the population by studying the long time behavior of the
solution of the Cauchy problem. We consider three sets of
assumptions on the growth function. In the so-called {\it confined
case} we determine a critical climate change speed for the
extinction or survival of the population, the latter case taking place by \lq\lq strictly following the climate shift''. In the so-called {\it
environmental gradient case}, or {\it unconfined case}, we additionally determine the propagation speed
of the population when it survives: thanks to a combination of migration and evolution, it can here be different from the speed of the climate shift. Finally, we consider {\it mixed scenarios}, that are complex situations, where the
growth function satisfies the conditions of the confined case on the right, and the conditions of the unconfined case on the left.
The main difficulty comes from the nonlocal competition term that prevents
the use of classical methods based on comparison arguments. This difficulty
is overcome thanks to estimates on the tails of the solution, and a careful application of the parabolic Harnack inequality.
59 citations
TL;DR: It is assumed here that the coefficients of the fast equation depend on time, so that the classical formulation of the averaging principle in terms of the invariant measure of theFast equation is no longer available, and the time-dependent evolution family of measures associated with thefast equation is introduced.
Abstract: We study the validity of an averaging principle for a slow-fast system of stochastic reaction-diffusion equations. We assume here that the coefficients of the fast equation depend on time, so that the classical formulation of the averaging principle in terms of the invariant measure of the fast equation is no longer available. As an alternative, we introduce the time-dependent evolution family of measures associated with the fast equation. Under the assumption that the coefficients in the fast equation are almost periodic, the evolution family of measures is almost periodic. This allows us to identify the appropriate averaged equation and prove the validity of the averaging limit.
59 citations
TL;DR: It is shown that for complex balanced systems without boundary equilibria, each trajectory converges exponentially fast to the unique complex balance equilibrium, and a constructive proof is proposed to explicitly estimate the rate of convergence in the special case of a cyclic reaction.
Abstract: The quantitative convergence to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks with mass action kinetics is studied using the so-called entropy method. In the first part of the paper, by deriving explicitly the entropy dissipation, we show that for complex balanced systems without boundary equilibria, each trajectory converges exponentially fast to the unique complex balance equilibrium. Moreover, a constructive proof is proposed to explicitly estimate the rate of convergence in the special case of a cyclic reaction. In the second part of the paper, complex balanced systems with boundary equilibria are considered. We focus on a specific case involving three chemical substances for which the boundary equilibrium is shown to be unstable in some sense, so that exponential convergence to the unique strictly positive equilibrium is recovered.
58 citations
TL;DR: The Bresse system is a valid model for arched beams which reduces to the classical Timoshenko system when the arch curvature is zero as mentioned in this paper, and it is shown that the Timoshenko model is a singular limit of the Brosse system as $\ell \to 0$.
Abstract: The Bresse system is a valid model for arched beams which reduces to the classical Timoshenko system when the arch curvature $\ell$ is zero. Our first result shows the Timoshenko system as a singular limit of the Bresse system as $\ell \to 0$. The remaining results are concerned with the long-time dynamics of Bresse systems. In a general framework, allowing nonlinear damping and forcing terms, we prove the existence of a smooth global attractor with finite fractal dimension and exponential attractors as well. We also compare the Bresse system with the Timoshenko system, in the sense of the upper-semicontinuity of their attractors as $\ell \to 0$.
53 citations
TL;DR: In this article, a splitting variant of the Jordan-Kinderlehrer-Otto scheme was proposed to handle gradient flows with respect to the Kantorovich-Fisher-Rao metric.
Abstract: In this article we set up a splitting variant of the Jordan--Kinderlehrer--Otto scheme in order to handle gradient flows with respect to the Kantorovich--Fisher--Rao metric, recently introduced and defined on the space of positive Radon measure with varying masses. We perform successively a time step for the quadratic Wasserstein/Monge--Kantorovich distance and then for the Hellinger/Fisher--Rao distance. Exploiting some inf-convolution structure of the metric we show convergence of the whole process for the standard class of energy functionals under suitable compactness assumptions and investigate in details the case of internal energies. The interest is double: On the one hand we prove existence of weak solutions for a certain class of reaction-advection-diffusion equations, and on the other hand this process is constructive and well adapted to available numerical solvers.
49 citations
TL;DR: In this article, a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs is presented. But this result requires an O(L) continuity (with respect to the path) for the semisolutions and for the nonlinearity defining the equation.
Abstract: We prove a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs. In contrast with the previous result in Ekren, Touzi, and Zhang [Ann. Probab., 44 (2016), pp. 2507--2553], our conditions are easier to check and allow for the degenerate case, thus including first order path-dependent PDEs. Our argument follows the regularization method as introduced by Jensen, Lions, and Souganidis [Proc. Amer. Math. Soc., 102, (1988)] in the corresponding finite-dimensional PDE setting. The present argument significantly simplifies the comparison proof of Ekren, Touzi, and Zhang but requires an $\mathbb{L}^p$-type of continuity (with respect to the path) for the viscosity semisolutions and for the nonlinearity defining the equation.
48 citations
TL;DR: The solution in this paper is exactly a Nishida--Smoller type solution due to the fact that initial energy $E_{0}$ can be large as long as $\gamma$ is close to 1 and $
u$ is suitably large.
Abstract: In this paper, we study the Cauchy problem of the isentropic compressible magnetohydrodynamic equations in $\mathbb{R}^{3}$. When $((\gamma-1)^{\frac{1}{9}}+
u^{-\frac{1}{4}})E_0$ is suitably small, a result on the existence of the global classical solution is established, where $\gamma$, $
u$, and $E_0$ represent the adiabatic exponent, resistivity coefficient, and initial energy, respectively. The solution in this paper is exactly a Nishida--Smoller type solution due to the fact that initial energy $E_{0}$ can be large as long as $\gamma$ is close to 1 and $
u$ is suitably large. Our result also improves the one established by [H. L. Li, X. Y. Xu, and J. W. Zhang, SIAM J. Math. Anal., 45 (2013), pp. 1356--1387], where with small initial energy but possibly large oscillations, the existence of the classical solution was proved.
TL;DR: It is shown that the problem of phase retrieval is never uniformly stable in infinite dimensions and the stability properties cannot be improved by oversampling the underlying discrete frame.
Abstract: We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main result states that also in this framework, the problem of phase retrieval is never uniformly stable in infinite dimensions. On the other hand, we show weak stability of the problem. This complements recent work by Cahill, Casazza, and Daubechies [Trans. Amer. Math. Soc. Ser. B, 3 (2016), pp. 63--76], where it has been shown that phase retrieval is always unstable for the setting of discrete frames in Hilbert spaces. In particular, our result implies that the stability properties cannot be improved by oversampling the underlying discrete frame. We generalize the notion of complement property (CP) to the setting of continuous frames for Banach spaces (over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$) and verify that it is a necessary condition for uniqueness of the p...
TL;DR: This work proves the global existence and uniqueness of weak solutions, with the initial data taken as small $L^\infty$ perturbations of functions in the space X, by generalizing in a uniform way the result on the uniqueness of strong solutions with continuous initial data.
Abstract: We establish some conditional uniqueness of weak solutions to the viscous primitive equations, and as an application, we prove the global existence and uniqueness of weak solutions, with the initial data taken as small $L^\infty$ perturbations of functions in the space $X=\{v\in (L^6(\Omega))^2|\partial_zv\in (L^2(\Omega))^2\}$; in particular, the initial data are allowed to be discontinuous. Our result generalizes in a uniform way the result on the uniqueness of weak solutions with continuous initial data and that of the so-called $z$-weak solutions.
TL;DR: In this article, Saut et al. provided a long time well-posedness result for a Boussinesq system with weakly nonlinear water wave propagation, which is a continuation of a previous work by two of the authors [J.-C. Saut and Li Xu, 2012].
Abstract: This paper is a continuation of a previous work by two of the authors [J.-C. Saut and Li Xu, J. Math. Pures Appl. (9), 97 (2012), pp. 635--662.] on long time existence for Boussinesq systems modeling the propagation of long, weakly nonlinear water waves. We provide proofs on examples not considered in the Sant and Xu paper; in particular, we prove a long time well-posedness result for a delicate “strongly dispersive” Boussinesq system.
TL;DR: In this paper, anomalous localized resonance has been used to cloaking an arbitrary object via localized resonance and provides their analysis in two and three dimensions, in which the cloaking device is independent of the object.
Abstract: In this paper, we present various schemes of cloaking an arbitrary object via anomalous localized resonance and provide their analysis in two and three dimensions. This is a way to cloak an object using negative index materials in which the cloaking device is independent of the object. As a result, we show that in the two dimensional quasi-static regime an annular plasmonic structure of coefficient $-1$ cloaks small but finite size objects nearby. We also discuss its connections with superlensing and cloaking using complementary media. In particular, we confirm the possibility that a lens can act like a cloak and conversely. This possibility was raised about a decade ago in the literature.
TL;DR: It is shown that the additional shear flow, if it is chosen sufficiently large, suppresses one dimension of the dynamics and hence can suppress blow-up.
Abstract: In this paper we consider the parabolic-elliptic Patlak--Keller--Segel models in ${\mathbb T}^d$ with d=2,3 with the additional effect of advection by a large shear flow. Without the shear flow, the model is $L^1$ critical in two dimensions with critical mass $8\pi$; solutions with mass less than $8\pi$ are global and solutions with mass larger than $8 \pi$ with finite second moment all blow up in finite time. In three dimensions, the model is $L^{3/2}$ critical and $L^1$ supercritical; there exist solutions with arbitrarily small mass which blow up in finite time arbitrarily fast. We show that the additional shear flow, if it is chosen sufficiently large, suppresses one dimension of the dynamics and hence can suppress blow-up. In two dimensions, the problem becomes effectively $L^1$ subcritical and so all solutions are global in time (if the shear flow is chosen large). In three dimensions, the problem is effectively $L^1$ critical, and solutions with mass less than $8\pi$ are global in time (and for all...
TL;DR: The theory of the principal eigenvalue is established for the eigen value problem associated with a linear time-periodic parabolic cooperative system with some zero diffusion coefficients.
Abstract: The theory of the principal eigenvalue is established for the eigenvalue problem associated with a linear time-periodic parabolic cooperative system with some zero diffusion coefficients. Then we apply it to a benthic-drift population model and obtain a threshold-type result on its global dynamics in terms of the basic reproduction number.
TL;DR: In this paper, the authors considered the homogenization problem for a nonlocal linear operator with a kernel of convolution type in a medium with a periodic structure and studied the limit behavior of the rescaled operators as the scaling parameter tends to 0.
Abstract: The paper deals with a homogenization problem for a nonlocal linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behavior of the rescaled operators as the scaling parameter tends to 0. More precisely we show that in the topology of resolvent convergence the family of rescaled operators converges to a second order elliptic operator with constant coefficients. We also prove the convergence of the corresponding semigroups both in $L^2$ space and the space of continuous functions and show that for the related family of Markov processes the invariance principle holds.
TL;DR: In this article, the authors consider ground states of two-dimensional Bose-Einstein condensates in a trap with attractive interactions, which can be described equivalently by positive minimizers of the Gross-Pitaevskii energy functional, and prove the local uniqueness and refined spike profiles of ground states as $a
earrow a^*$, provided that the trapping potential $h(x)$ is homogeneous and $H(y)=\int_{\mathbb{R}^2} h(x+y)w^2(x
Abstract: We consider ground states of two-dimensional Bose-Einstein condensates in a trap with attractive interactions, which can be described equivalently by positive minimizers of the $L^2$-critical constraint Gross-Pitaevskii energy functional. It is known that ground states exist if and only if $a< a^*:= \|w\|_2^2$, where $a$ denotes the interaction strength and $w$ is the unique positive solution of $\Delta w-w+w^3=0$ in ${\mathbb{R}}^2$. In this paper, we prove the local uniqueness and refined spike profiles of ground states as $a
earrow a^*$, provided that the trapping potential $h(x)$ is homogeneous and $H(y)=\int_{\mathbb{R}^2} h(x+y)w^2(x)dx$ admits a unique and nondegenerate critical point.
TL;DR: It is proved that functions of intrinsic-mode type (a classical models for signals) behave essentially like holomorphic functions: adding a pure carrier frequency ensures that the anti-holomorphic part is much smaller than the holomorphic part.
Abstract: We prove that functions of intrinsic-mode type (a classical models for signals) behave essentially like holomorphic functions: adding a pure carrier frequency $e^{int}$ ensures that the anti-holomorphic part is much smaller than the holomorphic part $ \| P_{-}(f)\|_{L^2} \ll \|P_{+}(f)\|_{L^2}.$ This enables us to use techniques from complex analysis, in particular the unwinding series. We study its stability and convergence properties and show that the unwinding series can provide a high-resolution, noise-robust time-frequency representation.
TL;DR: Stronger and more general versions of a trace theorem can be presented without imposing any extra regularity of the function in the interior of the domain other than being square integrable in a new nonlocal function space.
Abstract: It is a classical result of Sobolev spaces that any $H^1$ function has a well-defined $H^{1/2}$ trace on the boundary of a sufficient regular domain. In this work, we present stronger and more general versions of such a trace theorem in a new nonlocal function space ${\mathcal{S}}(\Omega)$ satisfying $H^1(\Omega)\subset {\mathcal{S}}(\Omega)\subset L^2(\Omega)$. The new space ${\mathcal{S}}(\Omega)$ is associated with a nonlocal norm characterized by a nonlocal interaction kernel that is defined heterogeneously with a special localization feature on the boundary. Through the heterogeneous localization, we are able to show that the $H^{1/2}$ norm of the trace on the boundary can be controlled by the nonlocal norm that is weaker than the classical $H^1$ norm. In fact, the trace theorems can be essentially shown without imposing any extra regularity of the function in the interior of the domain other than being square integrable. Implications of the new trace theorems for the coupling of local and nonlocal e...
TL;DR: The main aim of the paper is to study the convergence of the evolution of the empirical measure as n-to-infty, and renormalize the elastic energy to remove the potentially large self- or core energy.
Abstract: We consider systems of $n$ parallel edge dislocations in a single slip system, represented by points in a two-dimensional domain; the elastic medium is modeled as a continuum. We formulate the energy of this system in terms of the empirical measure of the dislocations and prove several convergence results in the limit $n\to\infty$. The main aim of the paper is to study the convergence of the evolution of the empirical measure as $n\to\infty$. We consider rate-independent, quasi-static evolutions, in which the motion of the dislocations is restricted to the same slip plane. This leads to a formulation of the quasi-static evolution problem in terms of a modified Wasserstein distance, which is only finite when the transport plan is slip-plane-confined. Since the focus is on interaction between dislocations, we renormalize the elastic energy to remove the potentially large self- or core energy. We prove Gamma-convergence of this renormalized energy, and we construct joint recovery sequences for which both the...
TL;DR: The critical strength of nonlinear compression is found, and it is proved that if the compression is stronger than this critical value, then singularity develops in finite time, and otherwise there are a class of initial data admitting global smooth solutions with maximum strength of compression equals to thiscritical value.
Abstract: It is well-known that singularity will develop in finite time for hyperbolic conservation laws from initial nonlinear compression no matter how small and smooth the data are. Classical results, including [P. Lax, J. Math. Phys., 5 (1964), pp. 611--614], [F. John, Comm. Pure Appl. Math., 27 (1974), pp. 377--405], [T. Liu, J. Differential Equations, 33 (1979), pp. 92--111], [T. Li, Y. Zhou, and D. Kong, Comm. Partial Differential Equations, 19 (1994), pp. 1263--1317], confirm that when initial data are small smooth perturbations near constant states, blowup in gradient of solutions occurs in finite time if initial data contain any compression in some truly nonlinear characteristic field, under some structural conditions. A natural question is, Will this picture keep true for large data problem of physical systems such as compressible Euler equations? One of the key issues is how to find an effective way to obtain sharp enough control on the density lower bound, which is known to decay to zero as time goes t...
TL;DR: In this paper, the uniqueness result for continuity equations with a velocity field whose derivative can be represented by a singular integral operator of an $L^1$ functal operator was established.
Abstract: In the first part of this paper we establish a uniqueness result for continuity equations with a velocity field whose derivative can be represented by a singular integral operator of an $L^1$ funct...
TL;DR: This paper proves the existence of a consistent quantum field theory that describes the Yukawa-like interaction of a non-relativistic nucleon field with a relativistic meson field in the classical limit by means of a Bohr correspondence principle.
Abstract: This paper studies the derivation of the nonlinear system of Schrodinger--Klein--Gordon (S-KG) equations, coupled by a Yukawa-type interaction, from a microscopic quantum field model of nonrelativi...
TL;DR: A positive answer is given to the question of the necessity of the same sufficient conditions in related Korn type inequalities for the full symmetric gradient, for negative Orlicz-Sobolev norms, and for the gradient of the Bogovskii operator.
Abstract: Necessary and sufficient conditions are exhibited for a Korn-type inequality to hold between (possibly different) Orlicz norms of the gradient of vector-valued functions and of the deviatoric part of their symmetric gradients. As a byproduct of our approach, a positive answer is given to the question of the necessity of the same sufficient conditions in related Korn-type inequalities for the full symmetric gradient, for negative Orlicz--Sobolev norms, and for the gradient of the Bogovskiǐ operator.
TL;DR: In this paper, the inviscid limit of the Navier-Stokes equations in a half-plane with Dirichlet boundary conditions was shown to hold in the energy norm.
Abstract: We consider the vanishing viscosity limit of the Navier--Stokes equations in a half-plane, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the product of the components of the Navier--Stokes solutions are equicontinuous at $x_2=0$. A sufficient condition for this to hold is that the tangential Navier--Stokes velocity remains uniformly bounded and has a uniformly integrable tangential gradient near the boundary.
TL;DR: This paper investigates the use of Stekloff eigenvalues for Maxwell's equations for the purpose of detecting changes in a scatterer using remote measurements of the scattered wave and proposes a modified Stkloff problem that restores compactness.
Abstract: In [F. Cakoni, D. Colton, S. Meng, and P. Monk, SIAM J. Appl. Math., 76 (2016), pp. 1737--1763] it was suggested to use Stekloff eigenvalues for the Helmholtz equation to detect changes in a scatterer using remote measurements of the scattered wave. This paper investigates the use of Stekloff eigenvalues for Maxwell's equations for the same purpose. Because the Stekloff eigenvalue problem for Maxwell's equations is not a standard eigenvalue problem for a compact operator, we propose a modified Stekloff problem that restores compactness. In order to measure the modified Stekloff eigenvalues of a domain from far field measurements we perturb the usual far field equation of the linear sampling method by using the far field pattern of an auxiliary impedance problem related to the modified Stekloff problem. We are then able to show (1) the existence of modified Stekloff eigenvalues and (2) the well-posedness of the corresponding auxiliary exterior impedance problem and (3) to provide theorems that support our ...
TL;DR: In this paper, the authors prove the existence of martingale solutions to the Navier-Stokes equations with respect to a set of weak solutions emanating from large inital data, set within a bounded domain.
Abstract: This article is devoted to the well-posedness of the stochastic compressible Navier--Stokes equations. We establish the global existence of an appropriate class of weak solutions emanating from large inital data, set within a bounded domain. The stochastic forcing is of multiplicative type, white in time and colored in space. Energy methods are used to merge techniques of P. L. Lions and E. Feireisl for the deterministic, compressible system with the theory of martingale solutions to the incompressible, stochastic system. Namely, we develop stochastic analogues of the weak compactness program of Lions, and use them to implement a martingale method. The existence proof involves four layers of approximating schemes. We combine the three layer scheme of Feireisl, Novotny, and Petzeltova for the deterministic, compressible system with a time splitting method used by Berthelin and Vovelle for the one dimensional stochastic compressible Euler equations.
TL;DR: In this paper, the existence of stationary solutions of a nonlocal aggregation equation with degenerate power-law diffusion and bounded attractive potential in arbitrary dimensions is investigated and compactness considerations are used to derive the existence for global minimizers of the corresponding energy.
Abstract: We investigate stationary solutions of a nonlocal aggregation equation with degenerate power-law diffusion and bounded attractive potential in arbitrary dimensions. Compact stationary solutions are characterized and compactness considerations are used to derive the existence of global minimizers of the corresponding energy depending on the prefactor of the degenerate diffusion for all exponents of the degenerate diffusion greater than one. We show that a global minimizer is compactly supported and, in case of quadratic diffusion, we prove that it is the unique stationary solution up to a translation. The existence of stationary solutions being only local minimizers is discussed.