gradient weighted estimates at the natural exponent for quasilinear parabolic equations abstract in this paper we obtain weighted norm inequalities for the spatial gradients of weak solutions to quasilinear parabolic equations with weights in the muckenhoupt class a q p r n 1 for q p on non smooth domains here the quasilinear nonlinearity is modeled after the standard p laplacian operator until now all the weighted estimates for the gradient were obtained only for exponents q p the results for exponents q p used the full complicated machinery of the calderon zygmund theory developed over the past few decades but the constants blow up as q p essentially because the maximal function is not bounded on l 1 in order to prove the weighted estimates for the gradient at the natural exponent i e q p we need to obtain improved a priori estimates below the natural exponent to this end we develop the technique of lipschitz truncation based on 3 25 and obtain significantly improved estimates below the natural exponent along the way we also obtain improved unweighted calderon zygmund type estimates below the natural exponent which is new even for the linear equations

non negative solutions of the evolution p laplacian equation initial traces and cauchy problem when 1 p 2 from the introduction we continue here the investigation initiated earlier by us trans amer math soc 314 1989 no 1 187 224 concerning the solvability of the cauchy problem and the existence of initial traces for nonnegative weak solutions of the nonlinear evolution equation 1 ut div du p 2 du 0 in st rn 0 t 0 2 both in terms of results and techniques the main difference stems from the fact that unlike solutions in the case p 2 solutions of 1 are not in general locally bounded

development of a formal likelihood function for improved bayesian inference of ephemeral catchments 1 the application of formal bayesian inferential approaches in hydrologic modeling is often criticized for requiring explicit assumptions to be made about the distribution of the errors via the likelihood function these assumptions can be adequate in some situations but often little attention is paid to the selection of an appropriate likelihood function this paper investigates the application of bayesian methods in modeling ephemeral catchments we consider two modeling case studies including a synthetically generated data set and a real data set of an arid australian catchment the case studies investigate some typical forms of the likelihood function which have been applied widely by previous researchers as well as introducing new implementations aimed at better addressing likelihood function assumptions in dry catchments the results of the case studies indicate the importance of explicitly accounting for model residuals that are highly positively skewed due to the presence of many zeros zero inflation arising from the dry spells experienced by the catchment specifically the form of the likelihood function was found to significantly impact the calibrated parameter posterior distributions which in turn have the potential to greatly affect the uncertainty estimates in each application the likelihood function that explicitly accounted for the nonconstant variance of the errors and the zero inflation of the errors resulted in 1 equivalent or better fits to the observed discharge in both timing and volume 2 superior estimation of the uncertainty as measured by the reliability and sharpness metrics and 3 more linear quantile quantile plots indicating the errors were more closely matched to the assumptions of this form of the likelihood function

improved parameter inference in catchment models 1 evaluating parameter uncertainty a bayesian methodology is developed to evaluate parameter uncertainty in catchment models fitted to a hydrologic response such as runoff the goal being to improve the chance of successful regionalization the catchment model is posed as a nonlinear regression model with stochastic errors possibly being both autocorrelated and heteroscedastic the end result of this methodology which may use box cox power transformations and arma error models is the posterior distribution which summarizes what is known about the catchment model parameters this can be simplified to a multivariate normal provided a linearization in parameter space is acceptable means of checking and improving this assumption are discussed the posterior standard deviations give a direct measure of parameter uncertainty and study of the posterior correlation matrix can indicate what kinds of data are required to improve the precision of poorly determined parameters finally a case study involving a nine parameter catchment model fitted to monthly runoff and soil moisture data is presented it is shown that use of ordinary least squares when its underlying error assumptions are violated gives an erroneous description of parameter uncertainty

time periodic solutions of quasilinear parabolic differential equations i dirichlet boundary conditions we study boundary value problems for quasilinear parabolic equations when the initial condition is replaced by periodicity in the time variable our approach is to relate the theory of such problems to the classical theory for initial boundary value problems in the process we generalize many previously known results

obstacle problem for a class of parabolic equations of generalized p laplacian type abstract we study nonlinear parabolic pdes with orlicz type growth conditions the main result gives the existence of a unique solution to the obstacle problem related to these equations to achieve this we show the boundedness of weak solutions and that a uniformly bounded sequence of weak supersolutions converges to a weak supersolution moreover we prove that if the obstacle is continuous so is the solution

global c regularity for second order quasilinear elliptic euqations in divergence form on considere des equations elliptiques quasilineaires d ordre 2 sous forme divergence sous des conditions aux limites de dirihlet

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

weighted orlicz estimates for general nonlinear parabolic equations over nonsmooth domains we introduce a parabolic analogue of muckenhoupt weights to study optimal weighted regularity in orlicz spaces for a general nonlinear parabolic problem of p laplacian type in divergence form over a nonsmooth domain assuming that the nonlinearity is measurable with respect to the time variable and has a small bounded mean oscillation bmo with respect to the spatial variables that the lateral boundary of the parabolic cylinder is reifenberg flat and that the associated weight belongs to a suitable parabolic muckenhoupt class we obtain a global gradient estimate for such a nonlinear parabolic problem by essentially proving that the gradient of the weak solution is as globally integrable as the nonhomogeneous term in the weighted orlicz space our results extend the existing regularity estimates in lebesgue spaces to weighted orlicz spaces

decreasing sequences of sigma fields and a measure change for brownian motion ii let f t t 0 be the filtration of a brownian motion b t t 0 on f p an example is given of a measure q p in the sense of absolute continuity for which f t t 0 is not the filtration of any brownian motion on f q this settles a 15 year old question

maximum likelihood bayesian averaging of spatial variability models in unsaturated fractured tuff 1 hydrologic analyses typically rely on a single conceptual mathematical model yet hydrologic environments are open and complex rendering them prone to multiple interpretations and mathematical descriptions adopting only one of these may lead to statistical bias and underestimation of uncertainty bayesian model averaging bma hoeting et al 1999 provides an optimal way to combine the predictions of several competing models and to assess their joint predictive uncertainty however it tends to be computationally demanding and relies heavily on prior information about model parameters neuman 2002 2003 proposed a maximum likelihood version mlbma of bma to render it computationally feasible and to allow dealing with cases where reliable prior information is lacking we apply mlbma to seven alternative variogram models of log air permeability data from single hole pneumatic injection tests in six boreholes at the apache leap research site alrs in central arizona unbiased ml estimates of variogram and drift parameters are obtained using adjoint state maximum likelihood cross validation samper and neuman 1989a in conjunction with universal kriging and generalized least squares standard information criteria provide an ambiguous ranking of the models which does not justify selecting one of them and discarding all others as is commonly done in practice instead we eliminate some of the models based on their negligibly small posterior probabilities and use the rest to project the measured log permeabilities by kriging onto a rock volume containing the six boreholes we then average these four projections and associated kriging variances using the posterior probability of each model as weight finally we cross validate the results by eliminating from consideration all data from one borehole at a time repeating the above process and comparing the predictive capability of mlbma with that of each individual model we find that mlbma is superior to any individual geostatistical model of log permeability among those we consider at the alrs

a shuffled complex evolution metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters author s vrugt ja gupta hv bouten w sorooshian s abstract markov chain monte carlo mcmc methods have become increasingly popular for estimating the posterior probability distribution of parameters in hydrologic models however mcmc methods require the a priori definition of a proposal or sampling distribution which determines the explorative capabilities and efficiency of the sampler and therefore the statistical properties of the markov chain and its rate of convergence in this paper we present an mcmc sampler entitled the shuffled complex evolution metropolis algorithm scem ua which is well suited to infer the posterior distribution of hydrologic model parameters the scem ua algorithm is a modified version of the original sce ua global optimization algorithm developed by duan et al 1992 the scem ua algorithm operates by merging the strengths of the metropolis algorithm controlled random search competitive evolution and complex shuffling in order to continuously update the proposal distribution and evolve the sampler to the posterior target distribution three case studies demonstrate that the adaptive capability of the scem ua algorithm significantly reduces the number of model simulations needed to infer the posterior distribution of the parameters when compared with the traditional metropolis hastings samplers

the problem of dirichlet for evolution one dimensional p laplacian with nonlinear source in the present paper we consider the dirichlet problem for one dimensional p laplacian with nonlinear source we obtain new a priori estimates of a solution and of the gradient of a solution and formulate conditions guaranteeing the global solvability of this problem our consideration includes singular case as well

riesz potentials and pointwise estimates of solutions to anisotropic porous medium equation abstract for an anisotropic porous medium equation with l 1 regular lower order terms we prove the local boundedness and continuity of weak solutions in terms of the linear riesz potential of the right hand side of equation

global gradient estimates for general nonlinear parabolic equations in nonsmooth domains we establish the natural calderon zygmund theory for a nonlinear parabolic equation of p laplacian type in divergence form 0 1 ut diva du x t div f p 2f in t by essentially proving that 0 2 f p lq t du p lq t for every q 1 the equation under consideration is of general type and not necessarily of variation form the involved nonlinearity a a x t is assumed to have a small bmo semi norm with respect to x t variables and the lateral boundary of the domain is assumed to be reifenberg flat as a consequence we are able to not only relax the known regularity requirements on the nonlinearity for such a regularity theory but also extend local results to a global one in a nonsmooth domain whose boundary has a fractal property we also find an optimal regularity estimate in orlicz sobolev spaces for such nonlinear parabolic problems

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

bounded solutions to a quasilinear and singular parabolic equation with p laplacian abstract in this paper we study the existence of a positive local in time solution for the following singular nonlinear problem with homogeneous dirichlet boundary conditions t u p u u f x u u in 0 t q t u 0 on 0 t u 0 in q t u 0 x u 0 0 in where stands for a regular bounded domain of r n p u is the p laplacian defined by p u div u p 2 u 2 p 0 and t 0 the nonlinear term f r r n r is a caratheodory function satisfying the growth condition f x s a s q 1 b c p p q for a a x s r and m where a c m 0 and b 0 are some constants and q p p where p p n n p if p n and p if p n we prove for any initial nonnegative data u 0 l r with r 2 large enough the existence of at least one weak solution to p in the case 2 1 p 1 we prove the uniqueness of the solution and further regularity results for that we use some estimates based on logarithmic sobolev inequalities to get ultracontractivity of a closely related semi group of linear operators

a markov chain monte carlo scheme for parameter estimation and inference in conceptual rainfall runoff modeling a fully bayesian approach to parameter estimation and inference in conceptual rainfall runoff models crrms is presented computations are performed using a markov chain monte carlo mcmc method based on the metropolis hastings algorithm single site and block updating schemes are used for model parameters subject to nonnegativity restrictions as well as interval equality and order constraints diagnostics for the convergence of the markov chain and crrm assessment are also considered the mcmc approach produces samples from the joint posterior distribution of the model parameters this provides more information than single point estimates and avoids the need to use a normal approximation to the posterior as the basis for inference the methodology is demonstrated using an eight parameter conceptual rainfall runoff model and two case studies from southeastern australia the first case study considers a watershed with high runoff yield over a 12 year period the second case study considers a watershed with low yield over a 17 year period the results indicate that 1 bayesian methods provide an objective framework for model criticism and choice 2 the proposed strategies for handling constraints on model parameters are effective 3 the model parameters are sensitive to likelihood function selection 4 the conventional approach of using a power transformation and an autoregressive process to stabilize error variance and model dependence in the residuals may have limited success and 5 some care is required in the implementation of the mcmc approach and reliable results will be difficult to obtain when crrm complexity exceeds the limitations of the rainfall runoff data at hand a key finding is that the mcmc scheme presented herein provides a powerful means of identifying specific inadequacies in the structure of crrms

pontryagin maximum principle for semilinear and quasilinearparabolic equations with pointwise state constraints this paper studies the first order necessary conditions for the optimal controls of semilinear and quasilinear parabolic partial differential equations with pointwise state constraints a pontryagin type maximum principle is obtained

higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth abstract the aim of this paper is to derive the self improving property of integrability for the spatial gradient of solutions to degenerate parabolic obstacle problem with irregular obstacles and p x t nonstandard growth more precisely we prove that the spatial gradient of the solution is integrable to a larger power than the natural one determined by the structural assumptions on the involved differential operator

stationary processes whose filtrations are standard we study the standard property of the natural filtration associated to a 0 1 valued stationary process in our main result we show that if the process has summable memory decay then the associated filtration is standard we prove it by coupling techniques for a process whose associated filtration is standard we construct a product type filtration extending it based upon the usual couplings and the vershik s criterion for standardness

global gradient estimates for the parabolic p x t laplacian equation abstract in this paper we consider the global gradient estimates for weak solutions of the parabolic p x t laplacian type equation in a reifenberg flat domain

boundary estimates for solutions of nonlinear degenerate parabolic systems on etablit la regularite partout jusqu a la frontiere pour des solutions faibles de systemes paraboliques du type u l t div du p 2 du l f l x t u du dans t l 1 2 m u t r m m n p max 1 2n n 2 est un ouvert borne de r n n 2 0 t t 0 t

bayesian analysis for uncertainty estimation of a canopy transpiration model 1 a bayesian approach was used to fit a conceptual transpiration model to half hourly transpiration rates for a sugar maple acer saccharum stand collected over a 5 month period and probabilistically estimate its parameter and prediction uncertainties the model used the penman monteith equation with the jarvis model for canopy conductance this deterministic model was extended by adding a normally distributed error term this extension enabled using markov chain monte carlo simulations to sample the posterior parameter distributions the residuals revealed approximate conformance to the assumption of normally distributed errors however minor systematic structures in the residuals at fine timescales suggested model changes that would potentially improve the modeling of transpiration results also indicated considerable uncertainties in the parameter and transpiration estimates this simple methodology of uncertainty analysis would facilitate the deductive step during the development cycle of deterministic conceptual models by accounting for these uncertainties while drawing inferences from data

the filtration of the split words process smorodinsky and laurent have initiated the study of the filtrations of split word processes in the framework of discrete negative time for these filtrations we show that laurent s sufficient condition for non standardness is also necessary thus yielding a practical standardness criterion in turn this criterion enables us to exhibit a non standard filtration which becomes standard when time is accelerated by omitting infinitely many instants of time

cauchy problem for fast diffusion equations with a gradient term we study the cauchy problem for fast diffusion equation with a gradient term of the form ut um duq where 0 0 and 0 0 however as q m the optimality of the stipulated assumptions is left open

holder estimates for parabolic p x t laplacian systems in this paper we establish the local holder continuity of the spatial gradient of weak solutions to the parabolic p x t laplacian system begin array lll partial t u rm div left a x t du p x t 2 du right 0 end array more precisely we prove that begin array lll du in c rm loc 0 alpha alpha 2 quad rm for some alpha in 0 1 end array provided p and a are holder continuous

nonuniqueness and multi bump solutions in parabolic problems with the p laplacian abstract the validity of the weak and strong comparison principles for degenerate parabolic partial differential equations with the p laplace operator p is investigated for p 2 this problem is reduced to the comparison of the trivial solution 0 by hypothesis with a nontrivial nonnegative solution u x t the problem is closely related also to the question of uniqueness of a nonnegative solution via the weak comparison principle in this article realistic counterexamples to the uniqueness of a nonnegative solution the weak comparison principle and the strong maximum principle are constructed with a nonsmooth reaction function that satisfies neither a lipschitz nor an osgood standard uniqueness condition nonnegative multi bump solutions with spatially disconnected compact supports and zero initial data are constructed between sub and supersolutions that have supports of the same type

hyperbolic phenomena in a strongly degenerate parabolic equation we consider the equation u t u u x x where 0 and where is a strictly increasing function with lim s we solve the associated cauchy problem for an increasing initial function and discuss to what extent the solution behaves qualitatively like solutions of the first order conservation law u t u x equations of this type arise for example in the theory of phase transitions where the corresponding free energy functional has a linear growth rate with respect to the gradient

parameter estimation of coupled water and energy balance models based on stationary constraints of surface states 1 we use a conditional averaging approach to estimate the parameters of a land surface water and energy balance model and then use the estimated parameters to partition net radiation into latent sensible and ground heat fluxes and precipitation into evapotranspiration and drainage plus runoff through conditional averaging of the modeled fluxes with respect to soil moisture and temperature we write an objective function that approximates the temperature and moisture dependent errors of the modeled fluxes in terms of atmospheric forcing e g precipitation and radiation surface states moisture s and temperature ts and model parameters the novelty of the approach is that the error term is estimated without comparison to measured fluxes instead it is inferred from the deviation of the conditionally averaged tendency terms expectation and expectation from zero since each of these terms equals zero in stationary systems but diverges from zero in the presence of misspecified parameters minimization of the approximated error yields parameters for model applications this strategy was previously studied for simple water balance models using soil moisture conditional averaging here we extend the idea to include energy balance fluxes and surface temperature conditioning the method is tested at two ameriflux sites vaira ranch california and kendall grassland arizona the estimated fluxes using only observed forcing and state variables are in reasonable agreement with field measurements because this method is based on conditional averages it can be applied to situations with subsampled or missing data that is continuous integration in time is not required

quantitative comparison of canopy conductance models using a bayesian approach 1 a quantitative model comparison methodology based on deviance information criterion a bayesian measure of the trade off between model complexity and goodness of fit is developed and demonstrated by comparing semiempirical transpiration models this methodology accounts for parameter and prediction uncertainties associated with such models and facilitates objective selection of the simplest model out of available alternatives which does not significantly compromise the ability to accurately model observations we use this methodology to compare various jarvis canopy conductance model configurations embedded within a larger transpiration model against canopy transpiration measured by sap flux the results indicate that descriptions of the dependence of stomatal conductance on vapor pressure deficit photosynthetic radiation and temperature as well as the gradual variation in canopy conductance through the season are essential in the transpiration model use of soil moisture was moderately significant but only when used with a hyperbolic vapor pressure deficit relationship subtle differences in model quality could be clearly associated with small structural changes through the use of this methodology the results also indicate that increments in model complexity are not always accompanied by improvements in model quality and that such improvements are conditional on model structure possible application of this methodology to compare complex semiempirical models of natural systems in general is also discussed

travelling waves in a convection diffusion equation abstract we study existence and stability of travelling waves for nonlinear convection diffusion equations in the 1 d euclidean space the diffusion coefficient depends on the gradient in analogy with the p laplacian and may be degenerate unconditional stability is established with respect to initial data perturbations in l 1 r

local calderon zygmund estimates for parabolic minimizers abstract we establish local calderon zygmund estimates for parabolic minimizers u to certain variational inequalities the formal corresponding parabolic partial differential equation would be t u div f z u d u u f z u d u div f p 2 f h with p 2 n n 2 however since we do not impose any condition on the existence of the derivative u f the pde above may not have any meaning at all for proving the calderon zygmund estimates we thus remain solely on the level of parabolic minimizers precisely for q p we prove that d u l loc q if u is locally holder continuous and f l loc q and h l loc q for a q depending on q

regularity of solutions of nonlinear degenerate parabolic systems soit le systeme parabolique u i t div u p 2 u i 0 i m avec p 1 ou u i u i x t grad x et x r n ouvert pour toute solution de ce systeme avec max 1 2n n 2 p u est continu dans t

regularity for the weak solutions to certain parabolic systems under certain growth condition abstract in this paper we consider the regularity of the weak solutions to a quasilinear parabolic systems which is a generalization of p laplacian of the type u t i a i u x f i x t u u i 1 n where the main part satisfies some ellipticity and f i satisfies certain growth conditions we prove boundedness of the solutions and the gradients of solutions to the systems by the means of the energy estimates and a nonlinear iteration procedure of the moser type in this paper