brezis gallouet wainger type inequalities and blow up criteria for navier stokes equations in unbounded domains we shall find the weakest norm that satisfies the brezis gallouet wainger type inequality under some conditions as an application of the brezis gallouet wainger type inequality we shall establish beale kato majda type blow up criteria of smooth solutions to the 3 d navier stokes equations in unbounded domains

an extension of the beale kato majda criterion for the euler equations the beale kato majda criterion asserts that smooth solutions to the euler equations remain bounded past t as long as is finite ohgr being the vorticity we show how to replace this by a weaker statement on where j is a frequency localization around

on the well posedness of the euler equations in the triebel lizorkin spaces we prove local in time unique existence and a blowup criterion for solutions in the triebel lizorkin space for the euler equations of inviscid incompressible fluid flows in n n 2 as a corollary we obtain global persistence of the initial regularity characterized by the triebel lizorkin spaces for the solutions of two dimensional euler equations to prove the results we establish the logarithmic inequality of the beale kato majda type the moser type of inequality as well as the commutator estimate in the triebel lizorkin spaces the key methods of proof used are the littlewood paley decomposition and the paradifferential calculus by j m bony 2002 john wiley sons inc

amplitude equations for a sub diffusive reaction diffusion system a sub diffusive reaction diffusion system with a positive definite memory operator and a nonlinear reaction term is analysed amplitude equations ginzburg landau type are derived for short wave turing and long wave hopf bifurcation points

systematic derivation of reaction diffusion equations with distributed delays and relations to fractional reaction diffusion equations and hyperbolic transport equations application to the theory of neolithic transition we introduce a general method for the systematic derivation of nonlinear reaction diffusion equations with distributed delays we study the interactions among different types of moving individuals atoms molecules quasiparticles biological organisms etc the motion of each species is described by the continuous time random walk theory analyzed in the literature for transport problems whereas the interactions among the species are described by a set of transformation rates which are nonlinear functions of the local concentrations of the different types of individuals we use the time interval between two jumps the transition time as an additional state variable and obtain a set of evolution equations which are local in time in order to make a connection with the transport models used in the literature we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so called generalized master equations the method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations hyperbolic reaction diffusion equations and delay differential reaction diffusion equations thus in the analysis of a given problem we can fit to the data the type of reaction diffusion equation and the corresponding physical and kinetic parameters the method is illustrated as a test case by the study of the neolithic transition we introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction diffusion equation for the population density we derive a delay evolution equation for the rate of advance of agriculture which illustrates an application of our analysis

turing instability in sub diffusive reaction diffusion systems linear stability characteristics of an anomalous sub diffusive activator inhibitor system are investigated through a mesoscopic model where anomaly ensues by a memory integro differential operator with an equal anomaly exponent for all species it is shown that monotonic instability known from normal diffusion persists in the anomalous system thereby allowing turing pattern occurrence presence of anomaly stabilizes the system by diminution of the range of diffusion coefficients ratio involving instability in addition the maximal growth rate is diminished proving the existence of an absolutely stable anomalous system

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

on the blow up of solutions of the 3 d euler equations in a bounded domain it is shown that if 0 hat t is the maximal interval of existence of a smooth solutionu of the incompressible euler equations in a bounded simply connected domain subseteq r3 then int 0 hat t left omega cdot t right l infty omega dt infty where u is the vorticity crucial to this result is a special estimate proven in of the maximum velocity gradient in terms of the maximum vorticity and a logarithmic term involving a higher norm of the vorticity

kinetic equations for reaction subdiffusion systems derivation and stability analysis we derive general kinetic equations for reacting and subdiffusing entities based on a nonlinear continuous time random walk formalism proposed by vlad and ross phys rev e 66 061908 2002 reaction and diffusion processes are separable in a typical reaction diffusion system and their combined influence on the evolution of the density of a species is a simple sum our derivation shows that this is no longer true for subdiffusive entities undergoing reactions the strong memory effects in the transport process i e the non markovian nature of subdiffusion results in a nontrivial combination of reactions and spatial dispersal which we discuss in detail we carry out a linear stability analysis of the derived reaction subdiffusion system to understand the effects of memory on pattern formation we find that the turing instability persists in the subdiffusive system however the memory modifies the turing threshold and the characteristics of the band of unstable modes close to this threshold

brezis gallouet wainger type inequality with a double logarithmic term in the holder space its sharp constants and extremal functions we investigate the sharp constants in a brezis gallouet wainger type inequality with a double logarithmic term in the holder space in a bounded domain in r n ibrahim majdoub and masmoudi gave the sharp constant in the 2 dimensional case we make precise estimates to give the sharp constants and pass to the case of higher dimensions n 2 we can also show that the inequality with fixed constants including the sharp ones admits an extremal function under a suitable condition when the domain is a ball 2000 mathematics subject classification primary 46e35 secondary 35j85

anomalous diffusion with linear reaction dynamics from continuous time random walks to fractional reaction diffusion equations we have revisited the problem of anomalously diffusing species modeled at the mesoscopic level using continuous time random walks to include linear reaction dynamics if a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term if the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term we have also developed further extensions of the ctrw model to include more general reaction dynamics

limiting case of the sobolev inequality in bmo with application to the euler equations we shall prove a logarithmic sobolev inequality by means of the bmo norm in the critical exponents as an application we shall establish a blow up criterion of solutions to the euler equations

an alternative proof of logarithmically improved beale kato majda type extension criteria for smooth solutions to the navier stokes equations abstract we give an alternative proof of a logarithmically improved beale kato majda type extension criterion for smooth solutions to the navier stokes equations in the whole space which was shown by fan et al 2011 by our method we can also establish a similar criterion to the above in case of the half space bounded domains and exterior domains

l2 decay for the navier stokes flow in halfspaces on etudie le comportement asymptotique dans l 2 des solutions faibles des equations de navier stokes dans des demi espaces

interior regularity criteria for suitable weak solutions of the navier stokes equations we present new interior regularity criteria for suitable weak solutions of the 3 d navier stokes equations a suitable weak solution is regular near an interior point z if either the scaled l p q x t norm of the velocity with 3 p 2 q 2 1 q or the l p q x t norm of the vorticity with 3 p 2 q 3 1 q or the l p q x t norm of the gradient of the vorticity with 3 p 2 q 4 1 q 1 p is sufficiently small near z

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

remarks on gagliardo nirenberg type inequality with critical sobolev space and bmo we consider the generalized gagliardo nirenberg inequality in mathbb r n in the homogeneous sobolev space dot h s r mathbb r n with the critical differential order s n r which describes the embedding such as l p mathbb r n cap dot h n r r mathbb r n subset l q mathbb r n for all q with p q n r which may be regarded as a generalization of the brezis gallouet wainger inequality

the critical sobolev inequalities in besov spaces and regularity criterion to some semi linear evolution equations we show the critical sobolev inequalities in the besov spaces with the logarithmic form such as brezis gallouet wainger and beale kato majda as an application of those inequalities the regularity problem under the critical condition to the navier stokes equations the euler equations in mathbb r n and the gradient flow to the harmonic map to the sphere are discussed namely the serrin ohyama type regularity criteria are improved in the terms of the besov spaces

lazer mckenna conjecture the critical case abstract we consider an elliptic problem of ambrosetti prodi type involving critical sobolev exponent on a bounded smooth domain of dimension six or higher by constructing solutions with many sharp peaks near the boundary of the domain but not on the boundary we prove that the number of solutions for this problem is unbounded as the parameter tends to infinity thereby proving the lazer mckenna conjecture in the critical case

l q l r estimates for solutions of the nonstationary stokes equations in an exterior domain and the navier stokes initial value problems inl q spaces

reaction subdiffusion equations to analyze possible generalizations of reaction diffusion schemes for the case of subdiffusion we discuss a simple monomolecular conversion a ensuremath rightarrow b we derive the corresponding kinetic equations for the local a and b concentrations their form is rather unusual the parameters of the reaction influence the diffusion term in the equation for a component a a consequence of the non markovian nature of subdiffusion the equation for the product contains a term which depends on the concentration of a at all previous times our discussion shows that reaction subdiffusion equations may not resemble the corresponding reaction diffusion ones and are not obtained by a trivial change of the diffusion operator for a subdiffusion one

brezis gallouet wainger type inequalities and a priori estimates of strong solutions to navier stokes equations abstract we shall find the largest normed space or the weakest norm that satisfies the brezis gallouet wainger type inequality as an application of such an inequality we shall establish the new a priori estimate of strong solutions to navier stokes equations which has an almost single exponential growth form with respect to the scaling invariant quantity of the vorticity our method is based on the double logarithmic type inequality by means of the morrey space

on blow up criteria of smooth solutions to the 3 d euler equations in a bounded domain abstract we show that a smooth solution of the 3 d euler equations in a bounded domain breaks down if and only if a certain norm of vorticity blows up at the same time here this norm is weaker than bmo norm

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists