Search

We study the well-posedness of evolution equation of the form u ′ (t) = A u (t) + C u (t) , t ≥ 0 , where A generates a C 0 -semigroup (T A (t)) t ≥ 0 with ‖ T A (t) ‖ ≤ M e ω t and C is a (possibly unbounded) linear operator in a Banach space X. We prove that if C satisfies D (A) ⊂ D (C) , ‖ C R (μ , A) ‖ ≤ K / (μ − ω) for each μ > ω , then, the equation is well-posed, i.e., A + C generates a C 0 -semigroup (T A + C (t)) t ≥ 0 satisfying ‖ T A + C (t) ‖ ≤ M e (ω + M K) t. Our approach is to use the Hille-Yosida's theorem. We consider the space of such operators C in X with norm ‖ C ‖ A : = 1 M sup μ > ω ⁡ ‖ (μ − ω) C R (μ , A) ‖ < ∞ , in which we show that the exponential dichotomy of (T A (t)) t ≥ 0 persists under small perturbation C. The obtained results seem to be new. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Hölder regularity for solutions of the nonlocal evolutionary equation ∂ t u + (− Δ p) s u = 0. Here, (− Δ p) s is the fractional p -Laplacian, 0 < s < 1 and 1 < p < 2. We establish Hölder regularity with explicit Hölder exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained Hölder exponents are almost sharp. Our results complement the previous results for the superquadratic case when p ≥ 2. [ABSTRACT FROM AUTHOR]

In this paper, we study the wave equation on infinite graphs. On one hand, in contrast to the wave equation on manifolds, we construct an example for the non-uniqueness for the Cauchy problem of the wave equation on graphs. On the other hand, we obtain a sharp uniqueness class for the solutions of the wave equation. The result follows from the time analyticity of the solutions to the wave equation in the uniqueness class. In the last part, we extend the result to a wide class of linear evolution equations. [ABSTRACT FROM AUTHOR]

In [19] , Smoczyk showed that parabolic expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we obtain representation formula for the corresponding linear hyperbolic evolution equation for the support function of locally convex curves of general nonzero rotation number, finding some natural similarities and differences in behaviour of solutions as compared with the parabolic case. As applications, we consider linear hyperbolic flows that preserve length or generalised enclosed area of the evolving curve, flow of curves with Neumann boundary conditions inside cones and hyperbolic approaches to the Yau problem of flowing one curve to another. We also consider radial graphs evolving by similar linear equations and linear higher order hyperbolic equations. Our results may be compared with the corresponding parabolic cases consider by the first author, Schrader and Wheeler, in [17]. [ABSTRACT FROM AUTHOR]

The current paper is devoted to the asymptotic behavior of a class of stochastic PDE. More precisely, with the help of the theory of integrated semigroups and a crucial estimate of the random Stieltjes convolution, we study the existence and smoothness of center-unstable invariant manifolds and center-stable foliations for a class of stochastic PDE with non-dense domain through the Lyapunov-Perron method. Finally, we give two examples, a stochastic age-structured model and a stochastic parabolic equation, to illustrate our results. [ABSTRACT FROM AUTHOR]

For the family of nonautonomous delay equations, we show that the generator of the evolution semigroup obtained from any such equation gives rise to an autonomous delay equation on a higher-dimensional space. We call it an induced delay equation. More significantly, we show that this equation can be used to study some important properties of the original dynamics in four main directions: the characterization of the hyperbolicity of the original delay equation via the hyperbolicity of the induced delay equation; the description of the spectral properties and of the corresponding Lyapunov exponents; the equivalence between the hyperbolicity of the induced delay equation and an admissibility property for its nonlinear perturbations; and, finally, the robustness of hyperbolicity under sufficiently small linear perturbations. The proofs of some of these results depend on a new variation of constants formula for the nonlinear perturbations of the induced delay equation that is also established in our work. [ABSTRACT FROM AUTHOR]

This paper concerns static and evolution equations involving degenerate curls. The topic is motivated by the mathematical problems of Meissner states of anisotropic superconductors, and of the nonlinear magneto-static Maxwell equations with strong anisotropy. Two types of degenerate curls are studied, among them one is the directional curl along a unit length vector field, which has been introduced by the author before, and the other is the matrix-curl, which is introduced in this paper. Because of the degeneracy produced by the degenerate curl of the unknown vector field in the leading term of the equation, the boundary condition should be prescribed on part of the boundary according to the nature of the degenerate curl. We formulate the Dirichelt type and natural type boundary conditions, and give existence of solutions both for the static equations and for the evolution equations. [ABSTRACT FROM AUTHOR]

In this article we provide new insights into the well-posedness and maximal regularity of systems of abstract evolution equations, in the framework of periodic Lebesgue spaces of vector-valued functions. Our abstract model is flexible enough as to admit time-fractional derivatives in the sense of Liouville-Grünwald. We characterize the maximal regularity property solely in terms of R -boundedness of a block operator-valued symbol, and provide corresponding estimates. In addition, we show practical criteria that imply the R -boundedness part of the characterization. We apply these criteria to show that the Keller-Segel system, as well as a reactor model system, have L q − L p maximal regularity. [ABSTRACT FROM AUTHOR]

In this paper we consider an abstract Cauchy problem for a Maxwell system modeling electromagnetic fields in the presence of an interface between optical media. The electric polarization is in general time-delayed and nonlinear, turning the macroscopic Maxwell equations into a system of nonlinear integro-differential equations. Within the framework of evolutionary equations, we obtain well-posedness in function spaces exponentially weighted in time and of different spatial regularity and formulate various conditions on the material functions, leading to exponential stability on a bounded domain. [ABSTRACT FROM AUTHOR]

In this article, we consider the plasma-vacuum interface problem for the incompressible ideal MHD. The plasma magnetic field is tangential to the interface while the vacuum magnetic field vanishes. We shall prove the stability of the interface under the Taylor sign condition. By deriving the evolution equation of the interface in the Eulerian coordinates, we are able to identify different stability mechanisms which correspond to the hyperboliticity of this evolution equation. Once the optimal regularity of the interface is obtained, all the other quantities can be estimated in the Eulerian coordinates. Therefore, we do not need the change of coordinates or the use of the Alinhac's good unknowns. [ABSTRACT FROM AUTHOR]

We study the global solvability, mean attractors and invariant measures for an abstract locally monotone and generally coercive SPDEs driven by infinite-dimensional superlinear noise defined in a dual space of intersection of finitely many Banach spaces. The main feature of this abstract system is that it covers a larger class of fundamental models which are included or not included previously. Under an extended locally monotone variational setting, we establish the global well-posedness, Itô's energy equality and existence of mean random attractors in some high-order Bochner spaces. The existence, uniqueness, support, (high-order and exponential) moment estimates, ergodicity, (pointwise and Wasserstein-type) exponentially mixing and asymptotic stability of invariant measures and evolution systems of measures are discussed for autonomous and nonautonomous stochastic equations. A stopping time technique is used to prove the convergence of solutions in probability in order to overcome the difficulty caused by the local monotonicity and superlinear growth of the coefficients. Our abstract results and unified methods are expected to be applied to various types of SPDEs like 2D Navier-Stokes equations, 2D MHD equations, 2D magnetic Bénard problem, Burgers type equations, 3D Leray α -model, convective Brinkman-Forchheimer equations, fractional (s , p) -Laplacian equations with monotone nonlinearities of polynomial growth of arbitrary order, and others. [ABSTRACT FROM AUTHOR]

We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a homogeneous diffusivity. Our results cover the singular stochastic p -Laplace equations and, more generally, singular stochastic Φ-Laplace equations with zero Dirichlet boundary conditions. We obtain improved moment estimates and quantitative convergence rates of the ergodic semigroup to the unique invariant measure, classified in a systematic way according to the degree of local degeneracy of the potential at the origin. We obtain new concentration results for the invariant measure and establish maximal dissipativity of the associated Kolmogorov operator. In particular, we recover the results for the curve shortening flow in the plane by Es-Sarhir et al. (2012) [22] , and improve the results by Liu and Tölle (2011) [41]. [ABSTRACT FROM AUTHOR]

We obtain necessary and sufficient conditions for the strongly L p well-posedness of three abstract evolution equations, arising from fractional Moore-Gibson-Thompson type equations which have recently appeared in the literature. We use Fourier multiplier techniques to derive new characterizations in terms of the R -boundedness of the operator-valued symbol associated to each abstract model, when endowed with the time-fractional Liouville-Grünwald derivative. As a consequence of our characterization, we give new insights into the differences between the models based on the structure of the respective operator-valued symbols and show novel applications by including several classes of operators other than the Laplacian. [ABSTRACT FROM AUTHOR]

This work is devoted to Poisson stable (including stationary, periodic, quasi-periodic, Bohr almost periodic, Bohr almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent) solutions for stochastic functional evolution equations (SFEEs) with infinite delay. First, we prove the existence of bounded mild solutions for SFEEs with infinite delay. Then, according to the relationship between the solution and (drift and diffusion) coefficients, we obtain such Poisson stable solutions. Because the solutions of the delay equations are not Markov, we employ the solution maps in some phase space as a viable alternative for studying further asymptotic properties, and we also discuss Poisson stable solution maps and their asymptotic stability. [ABSTRACT FROM AUTHOR]

In this article, we first verify the global well-posedness of the second order lattice systems with varying coefficients. Then we prove that the solution mappings form a continuous process on the time-dependent phase spaces and the process has a time-dependent pullback attractor. Afterwards, we establish that there exists a family of Borel probability measures carried by the time-dependent pullback attractor which possesses invariant property under the action of the process. Further, we formulate the definition of statistical solution for the addressed evolution equations on time-dependent phase spaces and prove its existence. Our results reveal that the statistical solution of the second order lattice systems with varying coefficients satisfies the Liouville theorem in Statistical Mechanics. Finally, we propose some interesting issues concerning the singular limiting behavior as the varying coefficients tend to zero. [ABSTRACT FROM AUTHOR]

We prove Schauder type estimates for solutions of stationary and evolution equations driven by weak generators of transition semigroups associated to a semilinear stochastic partial differential equations with values in a separable Hilbert space. [ABSTRACT FROM AUTHOR]

We consider a nonautonomous semilinear evolution problem that models some sort of propagation problem in nonlinear elastic rods and nonlinear ion-acoustic waves. We investigate the existence and stability of a family of pullback exponential attractors for our problem under suitable growth and dissipativeness conditions. Moreover, we also prove the upper and lower semicontinuity of this family of pullback exponential attractors at time zero. As a particular case, we obtain the existence of the pullback attractor in an appropriate space, we prove its upper semicontinuity and, lastly, we obtain a regularity result of this pullback attractor. [ABSTRACT FROM AUTHOR]

The Pontryagin type maximum principle and Bellman's dynamic programming principle serve as two of the most important tools in solving optimal control problems. There is a huge literature on the study of relationship between them. The main purpose of this paper is to investigate the relationship between the Pontryagin type maximum principle and the dynamic programming principle for control systems governed by stochastic evolution equations in infinite dimensional space, with the control variables entering into both the drift and the diffusion terms. To do so, we first prove the dynamic programming principle for those systems without employing the martingale solutions. Then we establish the desired relationships in both cases that value function associated is smooth or nonsmooth. For the nonsmooth case, in particular, by employing the relaxed transposition solution, we discover the connection between the superdifferentials and subdifferentials of value function and the first-order and second-order adjoint equations. [ABSTRACT FROM AUTHOR]

We study formation of singularities and persistence properties of solutions for a class of non-local and non-linear evolution equations with an arbitrary function and a non-negative (dissipation) parameter, which includes the famous Camassa-Holm equation and other recently reported hydrodynamic models as particular members. In case such a parameter is positive, solutions emanating from Cauchy problems have time decaying norms bounded from above by the norm of the corresponding initial datum. Whenever the function is even, for a given odd initial datum with slope satisfying a certain relation involving the dissipation parameter, then the corresponding solution of the problem breaks at finite time. We can also describe scenarios for the occurrence of wave breaking for more general initial data or functions, provided that those latter satisfy certain conditions on their first derivatives. We also study asymptotic behavior and unique continuation properties for solutions of the equation. [ABSTRACT FROM AUTHOR]

We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. Several properties of solutions are analyzed. Emphasis is placed on to what extent the surface in our setting qualitatively evolves similar as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but gives an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time. Additionally, a formal explanation of how our equations can be interpreted as a gradient flow is included. [ABSTRACT FROM AUTHOR]

In this paper we study the properties of a non-markovian version of the telegraph process, whose non-markovian character comes from a nonlocal in-time evolution equation that is satisfied by its probability density function. In the first part of the paper, using the theory of Volterra integral equations, we obtain an explicit formula for its moments, and we prove that the Carleman condition is satisfied. This shows that the distribution of the process is uniquely determined by its moments. We also obtain an explicit formula for the moment generating function. In the second part of the paper, we prove that the distribution of this process coincides with the distribution of a process of the form T (| W (t) |) where T (t) is the classical telegraph process, and | W (t) | is a random time whose distribution is related to a nonlocal in-time version of the wave equation. To this end, we construct the probability density function via subordination from the distribution of the classic telegraph process. Our results exhibit a strong interplay between this type of processes and subdiffusion theory. [ABSTRACT FROM AUTHOR]

We prove the existence of weak solutions to a system of two diffusion equations that are coupled by a pointwise volume constraint. The time evolution is given by gradient dynamics for a free energy functional. Our primary example is a model for the demixing of polymers, the corresponding energy is the one of Flory, Huggins and de Gennes. Due to the non-locality in the equations, the dynamics considered here is qualitatively different from the one found in the formally related Cahn-Hilliard equations. Our angle of attack is from the theory of optimal mass transport, that is, we consider the evolution equations for the two components as two gradient flows in the Wasserstein distance with one joint energy functional that has the volume constraint built in. The main difference to our previous work [6] is the nonlinearity of the energy density in the gradient part, which becomes singular at the interface between pure and mixed phases, and thus requires a more sophisticated analysis. [ABSTRACT FROM AUTHOR]

In [25] , Smoczyk showed that expansion of convex curves and hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article we consider L 2 (d θ) -gradient flows for a class of higher-order curvature functionals. These give rise to higher order linear parabolic equations for which we derive similar representation formulae for their solutions. Solutions exist for all time under natural conditions on the initial curve and converge exponentially fast in the smooth topology to multiply-covered circles. We consider both closed, embedded convex curves and closed, convex curves of higher rotation number. We give some corresponding remarks where relevant on open convex curves. We also consider corresponding 'globally constrained' flows which preserve the length or enclosed area of the evolving curve and a higher order approach to the Yau problem of evolving one convex planar curve to another. In an Appendix, we give some related second order results, including a version of the Yau problem for star-shaped curves. [ABSTRACT FROM AUTHOR]

We study the dependence of mild solutions to linear stochastic evolution equations on Hilbert space driven by Wiener noise, with drift having linear part of the type A + ε G , on the parameter ε. In particular, we study the limit and the asymptotic expansions in powers of ε of these solutions, as well as of functionals thereof, as ε → 0 , with good control on the remainder. These convergence and series expansion results are then applied to a parabolic perturbation of the Musiela SPDE of mathematical finance modeling the dynamics of forward rates. [ABSTRACT FROM AUTHOR]

In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish a local-in-time theory, i.e., existence, uniqueness and blow-up criterion for pathwise solutions in Sobolev spaces H s with s > 3. Thereafter, we give a picture of the long time behavior of the solutions based on the type of noise we consider. On one hand, we identify a family of noises such that blow-up can be prevented with probability 1, guaranteeing the existence and uniqueness of global solutions almost surely. On the other hand, in the particular linear noise case, we show that singularities occur in finite time with positive probability, and we derive lower bounds of these probabilities. To conclude, we introduce the notion of stability of exiting times and show that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data. [ABSTRACT FROM AUTHOR]

We present an analog of Massera Theorem for asymptotic periodic solutions of linear equations x ′ (t) = A (t) x (t) + f (t) , t ≥ 0 (⁎) , where the family of linear operators A (t) generates a 1-periodic process (U (t , s)) t ≥ s ≥ 0 in a Banach space X and f is asymptotic 1-periodic in the sense that lim t → ∞ ⁡ (f (t + 1) − f (t)) = 0. The main result says that if 1 is isolated in σ (U (1 , 0)) on the unit circle Γ, then (⁎) has an asymptotic 1-periodic mild solution if and only if it has an asymptotic mild solution that is bounded and uniformly continuous with precompact range. If 1 ∉ σ (U (1 , 0)) ∩ Γ , such an asymptotic 1-periodic mild solution always exists and unique within a function g (t) with lim t → ∞ ⁡ g (t) = 0. Our study relies on a spectral theory of functions on the half line and the evolution semigroups associated with linear equations. The obtained results seem to be new, even in the finite dimensional case. [ABSTRACT FROM AUTHOR]

Main purpose of this paper is to study the local propagation of singularities of viscosity solution to contact type evolutionary Hamilton-Jacobi equation D t u (t , x) + H (t , x , D x u (t , x) , u (t , x)) = 0. An important issue of this topic is the existence, uniqueness and regularity of the strict singular characteristic. We apply the recent existence and regularity results on the Herglotz' type variational problem to the aforementioned Hamilton-Jacobi equation with smooth initial data. We obtain some new results on the local structure of the cut set of the viscosity solution near non-conjugate singular points. Especially, we obtain an existence result of smooth strict singular characteristic from and to non-conjugate singular initial point based on the structure of the superdifferential of the solution, which is even new in the classical time-dependent case. We also get a global propagation result for the C 1 singular support in the contact case. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of multi-agent deterministic optimal control problems. We initially provide a thorough analysis of the Lagrangian, Eulerian and Kantorovich formulations of the problems, as well as of their relaxations. Then we exhibit some equivalence results among the various representations and compare the respective value functions. To do it, we combine techniques and ideas from optimal transportation, control theory, Young measures and evolution equations in Banach spaces. We further exploit the connections among Lagrangian and Eulerian descriptions to derive consistency results as the number of particles/agents tends to infinity. To that purpose we prove an empirical version of the Superposition Principle and obtain suitable Gamma-convergence results for the controlled systems. [ABSTRACT FROM AUTHOR]

The global in time existence of weak solutions to a cross-diffusion system with fractional diffusion in the whole space is proved. The equations describe the evolution of multi-species populations in the regime of large-distance interactions; they have been derived in the many-particle limit from moderately interacting particle systems with Lévy noise. The existence proof is based on a three-level approximation scheme, entropy and moment estimates, and a new Aubin–Lions compactness lemma in the whole space. [ABSTRACT FROM AUTHOR]

We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as particular cases, the Cauchy multipoint problem, the weighted mean value problem and the periodic problem. The dynamic is transformed into an abstract setting and by combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results. By the compactness of the semigroup generated by the linear operator, we do not assume any Lipschitzianity, nor compactness on the nonlinear term or on the nonlocal initial condition. In addition, the exploited approximation technique coupled to a Hartman-type inequality argument, allows to treat nonlinearities with superlinear growth. Moreover, regarding the periodic case, we are able to show the existence of at least one periodic solution on the half line. [ABSTRACT FROM AUTHOR]

We consider a class of linear second order differential equations with damping and external force. We investigate the link between a uniform bound on the forcing term and the corresponding ultimate bound on the velocity of solutions, and we study the dependence of that bound on the damping and on the "elastic force". We prove three results. First of all, in a rather general setting we show that different notions of bound are actually equivalent. Then we compute the optimal constants in the scalar case. Finally, we extend the results of the scalar case to abstract dissipative wave-type equations in Hilbert spaces. In that setting we obtain rather sharp estimates that are quite different from the scalar case, in both finite and infinite dimensional frameworks. The abstract theory applies, in particular, to dissipative wave, plate and beam equations. [ABSTRACT FROM AUTHOR]

In this work new global dynamic bifurcation theorems are established for local semiflows on complete metric spaces. These theorems are applied to the nonlinear evolution equation u t + A u = f λ (u) in a Banach space X , where A is a sectorial operator with compact resolvent, and f λ (0) = 0 for all λ ∈ R. In particular, if f λ takes the form f λ (u) = λ u + f (u) , it is shown that the global dynamic bifurcation branch of a bifurcation point (0 , λ 0) is necessarily unbounded without assuming the "crossing odd-multiplicity" condition. It has long been recognized that dynamical methods can be helpful in solving static problems. As an example, the bifurcation of elliptic equations on bounded domains associated with homogenous Dirichlet boundary condition is addressed. By considering the corresponding nonclassical parabolic flows and applying the global dynamic bifurcation theorems given here for local semiflows, some new results with global features are derived. [ABSTRACT FROM AUTHOR]

In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on T 1 × S 3. In the case of real-valued coefficients, we prove that an operator in this class is conjugated to a constant-coefficient operator satisfying a Diophantine condition, and that such conjugation preserves the global analytic hypoellipticity. In the case where the imaginary part of the coefficients is non-zero, we show that the operator is globally analytic hypoelliptic if the Nirenberg-Treves condition (P) holds, in addition to an analytic Diophantine condition. [ABSTRACT FROM AUTHOR]

In this paper we present an abstract maximal L p -regularity result up to T = ∞ , that is tuned to capture (linear) Partial Differential Equations of parabolic type, defined on a bounded domain and subject to finite dimensional, stabilizing, feedback controls acting on (a portion of) the boundary. Illustrations include, beside a more classical boundary parabolic example, two more recent settings: (i) the 3 d -Navier-Stokes equations with finite dimensional, localized, boundary tangential feedback stabilizing controls as well as Boussinesq systems with finite dimensional, localized, feedback, stabilizing, Dirichlet boundary control for the thermal equation. [ABSTRACT FROM AUTHOR]

We improve the decay argument by Bona and Li (1997) [5] for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady Degasperis-Procesi equation in the nonlocal formulation. In addition, we give a method which confirms the symmetry of solitary waves, including those of the maximum height. Finally, we discover how the symmetric structure is connected to the steady structure of solutions to the Degasperis-Procesi equation, and give a more intuitive proof for symmetric solutions to be traveling waves. The improved argument and new method above can be used for the decay rate of solitary waves to many other dispersive equations and will give new perspectives on symmetric solutions for general evolution equations. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a chemotaxis model of multiple sclerosis, which is proposed by Lombardo et al. (2017) [10]. The model consists of a chemotaxis quasi-linear parabolic equation describing the evolution of the density of activated macrophages, a semilinear parabolic equation ruling the evolution of the concentration of cytokine and an ordinary different equation modeling the evolution of the density of apoptotic oligodendrocytes. In this model, activated macrophages stimulated by a cytokine move along the concentration of cytokine gradient, which is termed chemotaxis. Our first purpose is to rigorously prove the boundedness and global existence of the classical solution to this model by applying heat operator semigroup theory and establishing some a priori estimates. The another purpose is to study the global asymptotic stability of a unique positive equilibrium, which together with the Turing instability are two aspects characteristic of mutual complement. [ABSTRACT FROM AUTHOR]

A non-local evolution equation of the Camassa-Holm type with dissipation is considered. The local well-posedness of the solutions of the Cauchy problem involving the equation is established via Kato's approach and the wave breaking scenario is also described. To prove such result, we firstly construct conserved currents for the equation and from them, conserved quantities. [ABSTRACT FROM AUTHOR]

In this paper we consider approximations of a class of third order linear evolution equations in time governed by fractional powers. We explicitly calculate the fractional powers of matricial operators associated with evolution equations of third order in time, and we characterize the partial scale of the fractional power of order spaces associated with these operators. As an application, we present parabolic approximations of the Moore-Gibson-Thompson type equations. [ABSTRACT FROM AUTHOR]

It is shown that the classical as well as quasilinear Keller-Segel systems with non-degenerate diffusion possess for given T -periodic and sufficiently small forcing functions a unique, strong T -time periodic solution. The proof given relies on the existence of strong T -periodic solutions for the linearized system, its characterization in terms of maximal L p -regularity of the underlying operator and a quasilinear version of the Arendt-Bu Theorem. The latter is of independent interest and yields the existence of strong T -periodic solutions to general quasilinear evolution equations under suitable conditions on the operators and the forcing terms. [ABSTRACT FROM AUTHOR]

In this article, we first prove, from the viewpoint of infinite dynamical system, sufficient conditions ensuring the existence of trajectory statistical solutions for autonomous evolution equations. Then we establish that the constructed trajectory statistical solutions possess invariant property and satisfy a Liouville type equation. Moreover, we reveal that the equation describing the invariant property of the trajectory statistical solutions is a particular situation of the Liouville type equation. Finally, we study the equations of three-dimensional incompressible magneto-micropolar fluids in detail and illustrate how to apply our abstract results to some concrete autonomous evolution equations. [ABSTRACT FROM AUTHOR]

In this work we study the evolution of the interface between two different fluids in two concentric cylinders when the velocity is given by the Navier-Stokes equation and one of the fluids is thin. We present a formal asymptotic derivation of the evolution equation for the interface under different scaling assumptions for the surface tension. We then study the different types of the stationary solutions and travelling waves for the resulting equation. In particular, we state a global well posedness result and using Center Manifold Theory, we obtain detailed information about the long time asymptotics of the solutions of the problem. [ABSTRACT FROM AUTHOR]

We analyze the sensitivity of linear quadratic optimal control problems governed by general evolution equations with bounded or admissible control operator. We show, that if the problem is stabilizable and detectable, the solution of the extremal equation can be bounded by the right-hand side including initial data with the bound being independent of the time horizon. Consequently, the influence of perturbations of the extremal equations decays exponentially in time. This property can for example be used to construct efficient space and time discretizations for a Model Predictive Control scheme. Furthermore, a turnpike property for unbounded but admissible control of general semigroups can be deduced. [ABSTRACT FROM AUTHOR]

This paper concerns the Liouville type result for the general semilinear elliptic equation (S) a i j (x) ∂ i j u (x) + K q i (x) ∂ i u (x) + f (x , u (x)) = 0 a.e. in R N , where f is of the KPP-monostable nonlinearity, as a continuation of the previous works of the second author [31,32]. The novelty of this work is that we allow f s (x , 0) to be sign-changing and to decay fast up to a Hardy potential near infinity. First, we introduce a weighted generalized principal eigenvalue and use it to characterize the Liouville type result for Eq. (S) that was proposed by H. Berestycki. Secondly, if (a i j) is the identity matrix and q is a compactly supported divergence-free vector field, we find a condition that Eq. (S) admits no positive solution for K > K ⋆ , where K ⋆ is a certain positive threshold. To achieve this, we derive some new techniques, thanks to the recent results on the principal spectral theory for elliptic operators [14] , to overcome some fundamental difficulties arising from the lack of compactness in the domain. This extends a nice result of Berestycki-Hamel-Nadirashvili [5] on the limit of eigenvalues with large drift to the case without periodic condition. Lastly, the well-posedness and long time behavior of the evolution equation corresponding to (S) are further investigated. The main tool of our work is based on the maximum principle for elliptic and parabolic equations however it is far from being obvious to see if the comparison principle for (S) holds or not. [ABSTRACT FROM AUTHOR]

We prove several results establishing existence and regularity of stable manifolds for different classes of special solutions for evolution equations (these equations may be ill-posed): a single specific solution, an invariant torus filled with quasiperiodic orbits or more general manifolds of solutions. In the later cases, which include several orbits, we also establish the invariant manifolds of an orbit depend smoothly on the orbit (analytically in the case of quasi-periodic orbits and finitely differentiably in the case of more general families). We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, analyticity) assumptions on the linearized equation, establishes the existence of the desired manifold. Related results appear in the literature, but our results allow that the nonlinearity is unbounded and we obtain smoothness of the invariant manifolds. This makes the results in this paper applicable to some several models of current interest that could not be treated otherwise. We discuss in detail the Boussinesq equation of water waves (similar phenomena happen in other long wave approximations) and complex Ginzburg-Landau equation. More recently, we observed [11] that our results also apply to Mean Field Games. Since the equations we consider may be ill-posed, part of the requirements for the stable manifold is that one can define the (forward) dynamics on them. Note also that the methods that are based in the existence of dynamics (such as graph transform) do not apply to ill-posed equation. We use the methods based on integral equations (Perron method) associated with the partial dynamics, but we need to take advantage of smoothing properties of the partial dynamics. Note that, even if the families of solutions we started with are finite dimensional, the stable manifolds may be infinite dimensional. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish first and second order necessary optimality conditions for optimal control problems of stochastic evolution equations with control and state constraints. The control acts both in the drift and diffusion terms and the control region is a nonempty closed subset of a separable Hilbert space. We employ some classical set-valued analysis tools and theories of the transposition solution of vector-valued backward stochastic evolution equations and the relaxed-transposition solution of operator-valued backward stochastic evolution equations to derive these optimality conditions. The correction part of the second order adjoint equation, which does not appear in the first order optimality condition, plays a fundamental role in the second order optimality condition. [ABSTRACT FROM AUTHOR]

We study the long-time behavior of solutions of the k -Hessian evolution equation u t = S k (D 2 u) , posed on a bounded domain of the n -dimensional space with homogeneous boundary conditions. To this end, we construct a separable solution and we show that the long-time behavior of u is precisely described by this special solution. Further, we initiate the study of that equation on the entire space, providing a new class of explicit and radially symmetric self-similar solutions that we call k - Barenblatt solutions. These solutions present some common properties as those of well-known Barenblatt solutions for the porous media equation and the p -Laplacian equation. [ABSTRACT FROM AUTHOR]

We study the generation of singularities from the initial datum for a solution of the Cauchy problem for a class of Hamilton-Jacobi equations of evolution. For such equations, we give conditions for the existence of singular generalized characteristics starting at the initial time from a given point of the domain, depending on the properties of the proximal subdifferential of the initial datum in a neighbourhood of that point. [ABSTRACT FROM AUTHOR]

We review the Lyapunov functional method for linear ODEs and give an explicit construction of such functionals that yields sharp decay estimates, including an extension to defective ODE systems. As an application, we consider three evolution equations, namely the linear convection-diffusion equation, the two velocity BGK model and the Fokker–Planck equation. Adding an uncertainty parameter to the equations and analyzing its linear sensitivity leads to defective ODE systems. By applying the Lyapunov functional construction, we prove sharp long time behavior of order (1 + t M) e − μ t , where M is the defect and μ is the spectral gap of the system. The appearance of the uncertainty parameter in the three applications makes it important to have decay estimates that are uniform in the non-defective limit. [ABSTRACT FROM AUTHOR]

We prove Schauder type estimates for stationary and evolution equations driven by the classical Ornstein-Uhlenbeck operator in a separable Banach space, endowed with a centered Gaussian measure. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Beris–Edwards system for incompressible nematic liquid crystal flows. The system under investigation consists of the Navier–Stokes equations for the fluid velocity u coupled with an evolution equation for the order parameter Q -tensor. One important feature of the system is that its elastic free energy takes a general form and in particular, it contains a cubic term that possibly makes it unbounded from below. In the two dimensional periodic setting, we prove that if the initial L ∞ -norm of the Q -tensor is properly small, then the system admits a unique global weak solution. The proof is based on the construction of a specific approximating system that preserves the L ∞ -norm of the Q -tensor along the time evolution. [ABSTRACT FROM AUTHOR]