global regular and singular solutions for a model of gravitating particles the existence of solutions of a nonlinear parabolic equation describing the gravitational interaction of particles is studied for the initial data in spaces of generalized pseudomeasures this approach permits us to relax regularity assumptions on the initial conditions and to prove asymptotic stability results for the above problem

the drift diffusion system in two dimensional critical hardy space abstract we establish the time local well posedness for large data of a solution to two dimensional drift diffusion system in the critical hardy space h 1 r 2 this solution is obtained in the hardy class for which the natural free energy is well defined from the initial time the time global well posedness result for small data is also obtained in a similar class the crucial parts of the proof are to employ the end point type of maximal regularity for the homogeneous heat equation and some new bilinear estimates in the hardy space

infinite energy solutions to the homogeneous boltzmann equation the goal of this work is to present an approach to the homogeneous boltzmann equation for maxwellian molecules with a physical collision kernel that allows us to construct unique solutions to the initial value problem in a space of probability measures defined via the fourier transform in that space the second moment of a measure is not assumed to be finite so infinite energy solutions are not a priori excluded from our considerations moreover we study the large time asymptotics of solutions and in a particular case we give an elementary proof of the asymptotic stability of self similar solutions obtained by a v bobylev and c cercignani j stat phys 106 2002 1039 1071 2009 wiley periodicals inc

a fisher kpp type equation with density dependent diffusion and convection travelling wave solutions this paper concerns processes described by a nonlinear partial differential equation that is an extension of the fisher and kpp equations including density dependent diffusion and nonlinear convection the set of wave speeds for which the equation admits a wavefront connecting its stable and unstable equilibrium states is characterized there is a minimal wave speed for this wave speed there is a unique wavefront which can be found explicitly it displays a sharp propagation front for all greater wave speeds there is a unique wavefront which does not possess this property for such waves the asymptotic behaviour as the equilibrium states are approached is determined

simulation of multi dimensional random fields by karhunen loeve expansion abstract the karhunen loeve k l expansion method is a powerful tool for simulating stationary and nonstationary gaussian and non gaussian stochastic processes with explicitly known covariance functions since the k l expansion requires the solution of fredholm integral equation of the second kind it is generally not feasible to simulate multi dimensional random fields this is because even the numerical solution of the fredholm integral multi dimensional eigenvalue problem is difficult to obtain in order to address this problem this paper develops a consistent generalization of k l expansion for multi dimensional random field simulation the new method decomposes an n dimensional random field into a total of n stochastic processes each can be represented by using the traditional k l expansion thus the developed method is embedded into the well established framework of the k l expansion for simulating stochastic process and obviating the need for solving the multi dimensional integral eigenvalue problems four examples including random fields with different kinds of covariance functions are used to demonstrate the application of the proposed method

stochastic cascades and 3 dimensional navier stokes equations in this article we study the incompressible navier stokes equations in 3 the non linear integral equation satisfied by the fourier transform of the laplacian of the velocity field can be interpreted in terms of a branching process and a composition rule along the associated tree we derive from this representation new classes where global existence and uniqueness can be proven

towards simultaneous reduction of both input and output spaces for interactive simulation based structural design abstract engineering design problems generally involve a high dimensional input space of design variables yielding an output space by means of costly high fidelity evaluations in order to decrease the overall cost reduced order models for the output space such as proper orthogonal decomposition pod and proper generalized decomposition pgd are an active area of research however little research has been conducted into alleviating the problems associated with a high dimensional input space in addition to higher dimensionality being an impediment to efficient design by itself complex shapes involve a high number of explicit implicit constraints restricting the design space geometric parameterization methods in traditional cad present difficulties in expressing these constraints leading to a high failure rate and the generation of inadmissible shapes in this paper we propose a simultaneous meta modeling protocol for both input and output spaces we perform a reparametrization of the input space using constrained shape interpolation by introducing the concept of an manifold of admissible meshed shapes the output space is reduced using constrained proper orthogonal decomposition by simultaneously using meta modeling for both spaces we facilitate interactive design space exploration for the purpose of design the proposed approach is applied to the industrial problem of designing a car engine intake port

sharp profiles in degenerate and doubly degenerate fisher kpp equations this paper investigates the effects of a degenerate diffusion term in reaction diffusion models ut d u ux x g u with fisher kpp type g both in the case when d 0 0 and when d 0 d 1 0 with d u 0 elsewhere we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c and we show the appearance of a sharp type profile when c c these results solve recent conjectures formulated by sanchez garduno and maini j differential equations 117 1995 281 and satnoianu et al discrete continuous dyn systems series b 1 2000 339

the vertex morphing method for node based shape optimization abstract the vertex morphing method a consistent surface control approach for shape optimization is presented this method introduces a control field and a map that relates it to the geometry field the optimization problem is solved in the control space and the map is defined such that both surface smoothness and mesh regularity criteria are satisfied simultaneously and synchronously in the geometry field the control field is discretized with the same discretization as the one of the geometry which leads to the richest design space possible for optimization of a discrete geometry the capabilities of the vertex morphing method are evaluated within an adjoint based cfd shape optimization workflow several engineering cases mainly from automotive industry are tested the method has shown significant success in performance efficiency and robustness even for problems with extreme geometrical complexities and millions of design parameters for applications in structural as well as fluid mechanics and their combination

a parameterized non intrusive reduced order model and error analysis for general time dependent nonlinear partial differential equations and its applications abstract a novel parameterized non intrusive reduced order model p nirom based on proper orthogonal decomposition pod has been developed this p nirom is a generic and efficient approach for model reduction of parameterized partial differential equations p pdes over existing parameterized reduced order models p rom most of them are based on the reduced basis method it is non intrusive and independent on partial differential equations and computational codes during the training process the smolyak sparse grid method is used to select a set of parameters over a specific parameterized space p r p for each selected parameter the reduced basis functions are generated from the snapshots derived from a run of the high fidelity model more generally the snapshots and basis function sets for any parameters over p can be obtained using an interpolation method the p nirom can then be constructed by using our recently developed technique xiao et al 2015 41 42 where either the smolyak or radial basis function rbf methods are used to generate a set of hyper surfaces representing the underlying dynamical system over the reduced space the new p nirom technique has been applied to parameterized navier stokes equations and implemented with an unstructured mesh finite element model the capability of this p nirom has been illustrated numerically by two test cases flow past a cylinder and lock exchange case the prediction capabilities of the p nirom have been evaluated by varying the viscosity initial and boundary conditions the results show that this p nirom has captured the quasi totality of the details of the flow with cpu speedup of three orders of magnitude an error analysis for the p nirom has been carried out

existence and uniqueness theorem on weak solutions to the parabolic elliptic keller segel system abstract in r n n 3 we first define a notion of weak solutions to the keller segel system of parabolic elliptic type in the scaling invariant class l s 0 t l r r n for 2 s n r 2 with n 2 r n any condition on derivatives of solutions is not required at all the local existence theorem of weak solutions is established for every initial data in l n 2 r n we prove also their uniqueness as for the marginal case when r n 2 we show that if n 4 then the class c 0 t l n 2 r n enables us to obtain the only weak solution

parametric solutions involving geometry a step towards efficient shape optimization abstract optimization of manufacturing processes or structures involves the optimal choice of many parameters process parameters material parameters or geometrical parameters usual strategies proceed by defining a trial choice of those parameters and then solving the resulting model then an appropriate cost function is evaluated and its optimality checked while the optimum is not reached the process parameters should be updated by using an appropriate optimization procedure and then the model must be solved again for the updated process parameters thus a direct numerical solution is needed for each choice of the process parameters with the subsequent impact on the computing time in this work we focus on shape optimization that involves the appropriate choice of some parameters defining the problem geometry the main objective of this work is to describe an original approach for computing an off line parametric solution that is a solution able to include information for different parameter values and also allowing to compute readily the sensitivities the curse of dimensionality is circumvented by invoking the proper generalized decomposition pgd introduced in former works which is applied here to compute geometrically parametrized solutions

end point maximal regularity and its application to two dimensional keller segel system we prove the maximal l regularity of the cauchy problem of the heat equation in the besov space dot b 1 rho 0 mathbb r n 1 which is not umd space and as its application we establish the time local well posedness of the solution of two dimensional nonlinear parabolic system with the poisson equation in dot b 1 2 0 mathbb r 2 where the equation is considered in the space invariant by a scaling and particularly the natural free energy is well defined from the initial time the small data global existence is also obtained in the same class

existence and uniqueness theorem on mild solutions to the keller segel system coupled with the navier stokes fluid abstract we consider the keller segel system coupled with the navier stokes fluid in the whole space and prove the existence of global mild solutions with the small initial data in the scaling invariant space our method is based on the implicit function theorem which yields necessarily continuous dependence of solutions for the initial data as a byproduct we show the asymptotic stability of solutions as the time goes to infinity since we may deal with the initial data in the weak l p spaces the existence of self similar solutions provided the initial data are small homogeneous functions

global well posedness for keller segel system in besov type spaces abstract the cauchy problems for keller segel system are studied using homogeneous besov spaces with the homogeneous besov spaces b p 2 n p r n which is the scaling critical case for keller segel system global solutions for small initial data are obtained in the space in addition ill posedness for keller segel system is also studied

scaling in nonlinear parabolic equations the cauchy problem for parabolic equations with quadratic nonlinearity is studied we investigate the existence of global in time solutions and their large time behavior assuming some scaling property of the equation as well as of the norm of the banach space in which the solutions are constructed some applications of these abstract results are indicated

convergence to travelling waves for quasilinear fisher kpp type equations we consider the cauchy problem ut u xx u t x r r u 0 x u0 x x r when the increasing function satisfies that 0 0 and the equation may degenerate at u 0 in the case of 0 0 we consider the case of u0 l r 0 u0 x 1 a e x r and the special case of u u u we prove that the solution approaches the travelling wave solution with speed c 1 spreading either to the right or to the left or to the two travelling waves moving in opposite directions

smooth or singular solutions to the navier stokes system abstract the existence of singular solutions of the incompressible navier stokes system with singular external forces the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions including stationary ones and a pointwise loss of smoothness for solutions are proved in the same function space of pseudomeasure type

travelling waves and finite propagation in a reaction diffusion equation abstract we study the existence of travelling wave solutions and the property of finite propagation for the reaction diffusion equation ut um xx un x t r 0 with m 1 0 n r and u u x t 0 we show that travelling waves exist globally only if m n 2 and only for velocities c c 2 m in the study of propagation we must take into account that solutions of the cauchy problem can be nonunique for n c 0 for m n 2 maximal solutions propagate instantly to the whole space if n

linear vs nonlinear selection for the propagation speed of the solutions of scalar reaction diffusion equations invading an unstable equilibrium we revisit the classical problem of speed selection for the propagation of disturbances in scalar reaction diffusion equations with one linearly stable and one linearly unstable equilibrium for a wide class of initial data this problem reduces to finding the minimal speed of the monotone traveling wave solutions connecting these two equilibria in one space dimension we introduce a variational characterization of these traveling wave solutions and give a necessary and sufficient condition for linear versus nonlinear selection mechanism we obtain sufficient conditions for the linear and nonlinear selection mechanisms that are easily verifiable our method also allows us to obtain efficient lower and upper bounds for the propagation speed c

shape optimization in ship hydrodynamics using computational fluid dynamics abstract a definite trend in computational applied mechanics is the development of integrated procedures for design optimization based on large scale numerical simulations simulation based design sbd in the present paper the fundamental elements of a sbd environment for shape optimization are presented and analyzed the focus is on complex engineering optimization problems which involve computationally highly expensive objective functions and nonlinear constraints advanced strategies adopted for reducing the overall computational effort are illustrated optimization algorithms for nonlinear programming problems are discussed as well as alternative techniques for shape variation and mesh manipulation necessary to automatically adapt the volume grid to the evolving shapes a new verification and validation v v methodology for assessing errors and uncertainties in simulation based optimization is also introduced based on the trends i e the differences between the numerically predicted improvement of the objective function and the actual improvement measured in a dedicated experimental campaign including consideration of numerical and experimental uncertainties two different sbd versions are then presented and demonstrated on a complex industrial problem namely the optimal shape redesign of a ship under real world geometrical and functional constraints whose evaluation during the optimization process involves repeated solutions of the reynolds averaged navier stokes equations finally an experimental campaign is carried out on the two optimized models to validate the computations and assess the success of the optimization process both the optimized models demonstrate improved characteristics beyond the numerical and experimental uncertainties confirming the validity of the sbd frameworks

existence and multiplicity results for a class of semilinear elliptic equations abstract we study the existence and multiplicity of nonnegative solutions as well as the behavior of corresponding parameter dependent branches to the equation u 1 u u m u n in a bounded domain r n endowed with the zero dirichlet boundary data where 0 m 1 and n 0 when 0 the obtained solutions can be seen as steady states of the corresponding reaction diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination in addition to the main new results we formulate a few relevant conjectures

large global in time solutions to a nonlocal model of chemotaxis abstract we consider the parabolic elliptic model for the chemotaxis with fractional anomalous diffusion global in time solutions are constructed under nearly optimal assumptions on the size of radial initial data moreover criteria for blowup of radial solutions in terms of suitable morrey spaces norms are derived

the basin of attraction of the steady states for a chemotaxis model in mathbb r 2 with critical mass we consider the cauchy problem for a parabolic elliptic system in mathbb r 2 which is amathematical model of chemotaxis and also amodel of self attracting particles in the critical mass case we determine the basin of attraction of the steady states for the cauchy problem through a lyapunov functional

design space dimensionality reduction in shape optimization by karhunen loeve expansion abstract the paper presents a methodology to reduce the dimension of design spaces in shape optimization problems while retaining a desired level of geometric variance the method is based on a generalized karhunen loeve expansion kle arbitrary shape modification spaces are assessed in terms of karhunen loeve modes eigenvectors and associated geometric variance eigenvalues the former are used as a basis in order to build a reduced dimensionality representation of the shape modification the method is demonstrated for the shape optimization of a high speed catamaran based on cfd simulations and aimed at the reduction of the wave component of calm water resistance kle is applied to three design spaces with large dimensionality 20 based on a free form deformation technique the space with the largest geometric variance is selected for dimensionality reduction and design optimization n dimensional design spaces are used with n 1 2 3 and 4 retaining up to the 95 of the geometric variance associated to the original space the correlation between the objective reduction achieved the dimension n and the geometric variance of the reduced dimensionality space is shown and found significant

traveling wave phenomena in some degenerate reaction diffusion equations abstract in this paper we study the existence of travelling wave solutions t w s u x t x ct for the equation formula g u where the reactive part g u is as in the fisher kpp equation and different assumptions are made on the non linear diffusion term d u both functions d and g are defined on the interval 0 1 the existence problem is analysed in the following two cases case 1 d 0 0 d u 0 u 0 1 d and g c 2 0 1 d 0 0 and d 0 0 we prove that if there exists a value of c c for which the equation possesses a travelling wave solution of sharp type it must be unique by using some continuity arguments we show that for 0 c c there are no t w s while for c c the equation has a continuum of t w s of front type the proof of uniqueness uses a monotonicity property of the solutions of a system of ordinary differential equations which is also proved case 2 d 0 d 0 0 d and g c 2 0 1 d 0 0 if in addition we impose d 0 0 with d u 0 u 0 1 we give sufficient conditions on c for the existence of t w s of front type meanwhile if d 0 d u u 0 1 we analyse just one example d u u 2 and g u u 1 u which has oscillatory t w s for 0 c 2 and t w s of front type for c 2 in both the above cases we use higher order terms in the taylor series and the centre manifold theorem in order to get the local behaviour around a non hyperbolic point of codimension one in the phase plane

largest well posed spaces for the general diffusion system with nonlocal interactions abstract the authors derive a general diffusion gd system with nonlocal interactions of special structure via energetic variational approach and observe that there exist two critical values of s i e s 1 2 1 for the nonlocal interactions where s 1 2 reflects how strong nonlocal property we have and s 1 affects the linearization and choice of initial data spaces the authors also establish the global existence and uniqueness of mild solution

the navier stokes equations in the weak l n space with time dependent external force abstract we consider the navier stokes equations with time dependent external force either in the whole time or in positive time with initial data with domain either the whole space the half space or an exterior domain of dimension n ge 3 we give conditions on the external force sufficient for the unique existence of small solutions in the weak l n space bounded for all time in particular this result gives sufficient conditions for the unique existence and the stability of small time periodic solutions or almost periodic solutions this result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak l n space with time independent external force

non trivial ω limit sets and oscillating solutions in a chemotaxis model in r2 with critical mass abstract this paper studies the cauchy problem for a parabolic elliptic system in r 2 modeling chemotaxis as well as self attracting particles in the critical mass case the fine dynamics of the model is ascertained in terms of the structure of the underlying limit sets according to the results of this paper any nonnegative radially symmetric bounded solution either stabilizes to a steady state as t or oscillates between two steady states moreover a rather general class of nonnegative initial data not necessarily radially symmetric for which the associated solutions exhibit a complex oscillatory behavior is constructed their limit sets consist of a nontrivial topological continuum of steady states besides the technical difficulties inherent to the lack of compactness of the resolvent operators one has to add the challenge that the problem is utterly non local consequently thought the basic ideas on the foundations of this paper might be considered classical most of the proofs throughout are extremely sophisticated and absolutely new in their full generality

survey of shape parameterization techniques for high fidelity multidisciplinary shape optimization a survey is provided of shape parameterization techniques formultidisciplinary optimization and some emerging ideas are highlighted the survey focuses on the suitability of available techniques for multidisciplinary applications of complex cone gurations using high e delity analysis tools such as computational e uid dynamics and computational structural mechanics the suitability criteria are based on the efe ciency effectiveness ease of implementation and availability of analytical sensitivities for geometry and grids a section on sensitivity analysis grid regeneration and grid deformation techniques is also provided