local existence and gevrey regularity of 3 d navier stokes equations with l p initial data abstract we obtain local existence and gevrey regularity of 3 d periodic navier stokes equations in case the sequence of fourier coefficients of the initial data is in l p p 3 2 the l p norm of the sequence of fourier coefficients of the solution and its analogous gevrey norm remains bounded on a time interval whose length depends only on the size of the body force and the l p norm of the fourier coefficient sequence of the initial data the control on the gevrey norm produces explicit estimates on the analyticity radius of the solution as in foias and temam j funct anal 87 1989 359 369 the results provide an alternate approach in estimating the space analyticity radius of solutions to navier stokes equations than the one presented by grujic and kukavica j funct anal 152 1998 447 466

identification of the initial population of a nonlinear predator prey system backwards in time abstract we study for the first time the ill posed backward problem for a contaminated nonlinear predator prey system whose velocities of migration depend on the total average populations in the considered space domain we propose a new regularized problem for which we are able to prove its unique solvability in theorem 1 moreover under some mild assumptions on the true solution we give useful and rigorous error estimates and convergence rates in both the l 2 and h 1 norms in theorem 2 theorem 3 respectively furthermore numerical simulations are performed to illustrate the accuracy and stability of the regularized solution

adams type inequalities and related elliptic partial differential equations in dimension four abstract motivated by ruf sani s recent work we prove an adams type inequality and a singular adams type inequality in the whole four dimensional euclidean space as applications of those inequalities a class of elliptic partial differential equations are considered existence of nontrivial weak solutions and multiplicity results are obtained via the mountain pass theorem and the ekeland s variational principle this is a continuation of our previous work about singular trudinger moser type inequality

global compactness properties of semilinear elliptic equations with critical exponential growth abstract sequences of positive solutions to semilinear elliptic equations of critical exponential growth in the plane either are precompact in the sobolev h 1 topology or concentrate at isolated points of the domain for energies allowing at most single point blow up we establish a universal blow up pattern near the concentration point and uniquely characterize the blow up energy in terms of a geometric limiting problem

gevrey regularity for the navier stokes in a half space abstract we consider the navier stokes equations posed on the half space with dirichlet boundary conditions we give a direct energy based proof for the instantaneous space time analyticity and gevrey class regularity of the solution uniformly up to the boundary of the half space

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

a return mapping algorithm for plane stress elastoplasticity an unconditionally stable algorithm for plane stress elastoplasticity is developed based upon the notion of elastic predictor return mapping plastic corrector enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of newton methods the accuracy of the algorithm is assessed by means of iso error maps the excellent performance of the algorithm for large time steps is illustrated in numerical experiments

new solutions for trudinger moser critical equations in r2 abstract let be a bounded smooth domain in r 2 we consider critical points of the trudinger moser type functional j u 1 2 u 2 2 e u 2 in h 0 1 namely solutions of the boundary value problem u u e u 2 0 with homogeneous dirichlet boundary conditions where 0 is a small parameter given k 1 we find conditions under which there exists a solution u which blows up at exactly k points in as 0 and j u 2 k we find that at least one such solution always exists if k 2 and is not simply connected if has d 1 holes in addition d 1 bubbling solutions with k 1 exist these results are existence counterparts of one by druet in o druet multibump analysis in dimension 2 quantification of blow up levels duke math j 132 2 2006 217 269 which classifies asymptotic bounded energy levels of blow up solutions for a class of nonlinearities of critical exponential growth including this one as a prototype case

on the domain of analyticity for solutions of second order analytic nonlinear differential equations abstract the radius of analyticity of periodic analytic functions can be characterized by the decay of their fourier coefficients this observation has led to the use of so called gevrey norms as a simple way of estimating the time evolution of the spatial radius of analyticity of solutions to parabolic as well as non parabolic partial differential equations in this paper we demonstrate using a simple explicitly solvable model equation that estimates on the radius of analyticity obtained by the usual gevrey class approach do not scale optimally across a family of solutions nor do they scale optimally as a function of the physical parameters of the equation we attribute the observed lack of sharpness to a specific embedding inequality and give a modified definition of the gevrey norms which is shown to finally yield a sharp estimate on the radius of analyticity

space analyticity for the navier stokes and related equations with initial data inlp abstract we introduce a method of estimating the space analyticity radius of solutions for the navier stokes and related equations in terms of l p and l norms of the initial data the method enables us to express the space analyticity radius for 3d navier stokes equations in terms of the reynolds number of the flow also for the kuramoto sivashinsky equation we give a partial answer to a conjecture that the radius of space analyticity on the attractor is independent of the spatial period

on a singular and nonhomogeneous n laplacian equation involving critical growth abstract in this paper we apply minimax methods to obtain existence and multiplicity of weak solutions for singular and nonhomogeneous elliptic equation of the form n u f x u x a h x in where u w 0 1 n n u div u n 2 u is the n laplacian a 0 n is a smooth bounded domain in r n n 2 containing the origin and h w 0 1 n w 1 n is a small perturbation h 0 motivated by a singular trudinger moser inequality we study the case when f x s has the maximal growth on s which allows to treat this problem variationally in the sobolev space w 0 1 n

remark on the rate of decay of higher order derivatives for solutions to the navier stokes equations in rn we present a new derivation of upper bounds for the decay of higher order derivatives of solutions to the unforced navier stokes equations in rn the method based on so called gevrey estimates also yields explicit bounds on the growth of the radius of analyticity of the solution in time moreover under the assumption that the navier stokes solution stays sufficiently close to a solution of the heat equation in the l2 norm a result known to be true for a large class of initial data lower bounds on the decay of higher order derivatives can be obtained

existence of analytic solutions for the classical stefan problem we prove that under mild regularity assumptions on the initial data the two phase classical stefan problem admits a unique solution that is analytic in space and time

local analyticity radii of solutions to the 3d navier stokes equations with locally analytic forcing abstract we introduce a new method for establishing local analyticity and estimating the local analyticity radius of a solutions to the 3d navier stokes equations at interior points the approach is based on rephrasing the problem in terms of second order parabolic systems which are then estimated using the mild solution approach the estimates agree with the global analyticity radius from 16 up to a logarithm

phenomena of critical exponent in ℝ2 in this paper we make an attempt to explain the critical phenomena in r 2 we do this by exhibiting a class of functions having e u 2 growth and for which u u in b r u 0 u 0 on b r do not admit a solution when r is sufficiently small where b r denotes the ball of radius r in r 2

sharp adams type inequalities in sobolev spaces wm nm rn for arbitrary integer m the main purpose of our paper is to prove sharp adams type inequalities in unbounded domains of rn for the sobolev space wm nm rn for any positive integer m less than n our results complement those of ruf and sani in press 35 where such inequalities have been established for even integer m we extend the main techniques of ruf and sani in press 35 which are the combinations of the comparison principle of talenti 1976 36 and trombetti and vazquez 1985 38 for polyharmonic operators and a symmetrization argument together with constructions of radial auxiliary functions to the case when m is odd moreover we offer a completely different but much simpler approach to prove the comparison principle using the power of bessel potentials and the riesz rearrangement see remarks 3 2 and 3 3 this approach is of independent interest and works for any differential operators with appropriate radial kernels as corollaries of our main theorems we will derive the adams type inequalities in the case when n 2m for all positive integer m by using different sobolev norms

time analyticity for the heat equation and navier stokes equations abstract we prove the analyticity in time for solutions of two parabolic equations in the whole space without any decaying or vanishing conditions one of them involves solutions to the heat equation of exponential growth of order 2 on m here m is r d or a complete noncompact manifold with ricci curvature bounded from below by a constant an implication is a sharp solvability condition for the cauchy problem of the backward heat equation which is a well known ill posed problem another implication is a sharp criteria for time analyticity of solutions down to the initial time the other pertains bounded mild solutions of the incompressible navier stokes equations in the whole space there are many long established analyticity results for the navier stokes equations see for example 17 and 10 for spatial and time analyticity in certain integral sense 4 for pointwise space time analyticity of 3 dimensional solutions to the cauchy problem and also the pointwise time analyticity results of 22 and 12 under zero boundary condition our result seems to be the first general pointwise time analyticity result for the cauchy problem for all dimensions whose proof involves only real variable method the proof involves a method of algebraically manipulating the integral kernel which appears applicable to other evolution equations

existence and non existence of radial solutions for elliptic equations with critical exponent in ℝ2 where 2 2 n n 2 is the critical sobolev exponent for the embedding h 52 c l2 0 criticality and subcriticality play important rbles concerning the solvability of equation 1 1 using by now classical variational methods one sees that equation 1 1 is solvable i f f has subcritical growth and satisfies some additional hypotheses while for f u iuip u the well known pohoiaev identity shows that the solvability of equation 1 1 is lost if p reaches 2 and r is starshaped in their important paper 6 brezis nirenberg showed that for n e 4 solvability is regained if one adds a lower order linear term to the critical term more precisely they showed that for f r ar lr12 2r one has a denotes the 1st eigenvalue of a on ha r

analyticity of solutions for a generalized euler equation abstract we consider the so called lake and great lake equations which are shallow water equations that describe the long time motion of an inviscid incompressible fluid contained in a shallow basin with a slowly spatially varying bottom a free upper surface and vertical side walls under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude if these equations are posed on a space periodic domain and the initial data are real analytic the solution remains real analytic for all times the proof is based on a characterization of gevrey classes in terms of decay of fourier coefficients in particular our result recovers known results for the euler equations in two and three spatial dimensions we believe the proof is new

bubbling solutions for moser trudinger type equations on compact riemann surfaces abstract we study an elliptic equation related to the moser trudinger inequality on a compact riemann surface s g g u u e u 2 1 s s u e u 2 d v g 0 in s s u d v g 0 where 0 is a small parameter s is the area of s g is the laplace beltrami operator and d v g is the area element given any integer k 1 under general conditions on s we find a bubbling solution u which blows up at exactly k points in s as 0 when s is a flat two torus in rectangular form we find that either seven or nine families of such solutions do exist for k 2 in particular in any square flat two torus actually nine families of bubbling solutions with two bubbling points do exist if s is a riemann surface with non constant robin s function then at least two bubbling solutions with k 1 exists

space analyticity for the nonlinear heat equation in a bounded domain 1 introductionin fot foias and temam introduced a method for estimating thespace analyticity radius of solutions of the navier stokes equation withperiodic boundary conditions for the numerous applications seee g cees crt dth ft gr k lo in our previous paper gk we did not use the fourier series and were thus able to treat the navier stokes and other semilinear parabolic equations with singular l