c1 α isometric embeddings of polar caps abstract we study isometric embeddings of c 2 riemannian manifolds in the euclidean space and we establish that the holder space c 1 1 2 is critical in a suitable sense in particular we prove that for 1 2 the levi civita connection of any isometric immersion is induced by the euclidean connection whereas for any 1 2 we construct c 1 isometric embeddings of portions of the standard 2 dimensional sphere for which such property fails

some remark on the existence of infinitely many nonphysical solutions to the incompressible navier stokes equations abstract we prove that there exist infinitely many distributional solutions with infinite kinetic energy to both the incompressible navier stokes equations in r 2 and burgers equation in r with vanishing initial data

the role of bkm type theorems in 3d euler navier stokes and cahn hilliard navier stokes analysis abstract the beale kato majda theorem contains a single criterion that controls the behaviour of solutions of the 3 d incompressible euler equations versions of this theorem are discussed in terms of the regularity issues surrounding the 3 d incompressible euler and navier stokes equations together with a phase field model for the statistical mechanics of binary mixtures called the 3 d cahn hilliard navier stokes chns equations a theorem of bkm type is established for the chns equations for the full parameter range moreover for this latter set it is shown that there exists a reynolds number and a bound on the energy dissipation rate that remarkably reproduces the r e 3 4 upper bound on the inverse kolmogorov length normally associated with the navier stokes equations alone an alternative length scale is introduced and discussed together with a set of pseudo spectral computations on a 12 8 3 grid

the two dimensional euler equations on singular exterior domains this paper is a follow up of gerard varet and lacave arch ration mech anal 209 1 131 170 2013 on the existence of global weak solutions to the two dimensional euler equations in singular domains in gerard varet and lacave arch ration mech anal 209 1 131 170 2013 we have established the existence of weak solutions for a large class of bounded domains with initial vorticity in lp p 1 for unbounded domains we have proved a similar result only when the initial vorticity is in l p c p 2 and when the domain is the exterior of a single obstacle the goal here is to retrieve these two restrictions we consider general initial vorticity in l 1 cap l p p 1 outside an arbitrary number of obstacles not reduced to points

non uniqueness of admissible weak solutions to compressible euler systems with source terms abstract we consider admissible weak solutions to the compressible euler system with source terms which include rotating shallow water system and the euler system with damping as special examples in the case of anti symmetric sources such as rotations for general piecewise lipschitz initial densities and some suitably constructed initial momentum we obtain infinitely many global admissible weak solutions furthermore we construct a class of finite states admissible weak solutions to the euler system with anti symmetric sources under the additional smallness assumption on the initial densities we also obtain multiple global in time admissible weak solutions for more general sources including damping the basic framework are based on the convex integration method developed by de lellis and szekelyhidi 13 14 for the euler system one of the main ingredients of this paper is the construction of specified localized plane wave perturbations which are compatible with a given source term

global ill posedness of the isentropic system of gas dynamics we consider the isentropic compressible euler system in 2 space dimensions with pressure law p 2 and we show the existence of classical riemann data i e pure jump discontinuities across a line for which there are infinitely many admissible bounded weak solutions bounded away from the void we also show that some of these riemann data are generated by a 1 dimensional compression wave our theorem leads therefore to lipschitz initial data for which there are infinitely many global bounded admissible weak solutions c 2015 wiley periodicals inc

kolmogorov s theory of turbulence and inviscid limit of the navier stokes equations in mathbb r 3 we are concerned with the inviscid limit of the navier stokes equations to the euler equations in mathbb r 3 we first observe that a pathwise kolmogorov hypothesis implies the uniform boundedness of the th order fractional derivatives of the velocity for some 0 in the space variables in l2 which is independent of the viscosity 0 then it is shown that this key observation yields the l2 equicontinuity in the time variable and the uniform bound in lq for some q 2 of the velocity independent of 0 these results lead to the strong convergence of solutions of the navier stokes equations to a solution of the euler equations in mathbb r 3 we also consider passive scalars coupled to the incompressible navier stokes equations and in this case find the weak star convergence for the passive scalars with a limit in the form of a young measure pdf depending on space and time not only do we offer a framework for mathematical existence theories but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place with the bounds that are independent of 0 that is in the high reynolds number limit

monotone riemannian metrics and relative entropy on noncommutative probability spaces we use the relative modular operator to define a generalized relative entropy for any convex operator function g on 0 satisfying g 1 0 we show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy a unique family parametrized by density matrices of continuous monotone riemannian metrics a unique geodesic distance on the space of density matrices and a unique monotone operator function satisfying certain symmetry and normalization conditions we describe these objects explicitly in several important special cases including g w log w which yields the familiar logarithmic relative entropy the relative entropies riemannian metrics and geodesic distances obtained by our procedure all contract under completely positive trace preserving maps we then define and study the maximal contraction associated with these quantities

general existence theorems for hamilton jacobi equations in the scalar and vectorial cases keywords dirichlet problem hamilton jacobi equation existence of solutions reference caa article 1997 002doi 10 1007 bf02392708view record in web of science record created on 2008 11 25 modified on 2017 05 12

onsager s conjecture on the energy conservation for solutions of euler s equation we give a simple proof of a result conjectured by onsager 1 on energy conservation for weak solutions of euler s equation

on admissibility criteria for weak solutions of the euler equations we consider solutions to the cauchy problem for the incompressible euler equations satisfying several additional requirements like the global and local energy inequalities using some techniques introduced in an earlier paper we show that for some bounded compactly supported initial data none of these admissibility criteria singles out a unique weak solution as a byproduct in more than one space dimension we show bounded initial data for which admissible solutions to the p system of isentropic gas dynamics in eulerian coordinates are not unique

dissipative continuous euler flows we show the existence of continuous periodic solutions of the 3d incompressible euler equations which dissipate the total kinetic energy

jointly convex quantum jensen divergences abstract we investigate the quantum jensen divergences from the viewpoint of joint convexity it turns out that the set of the functions which generate jointly convex quantum jensen divergences on positive matrices coincides with the matrix entropy class which has been introduced by chen and tropp quite recently 3

high dimensionality and h principle in pde in this note we present an analyst s point of view on the nash kuiper theorem and in particular highlight the very close connection to turbulence a paradigm example of a high dimensional phenomenon our aim is to explain recent applications of nash s ideas in connection with the incompressible euler equations and onsager s famous conjecture on the energy dissipation in 3d turbulence

dissipative euler flows with onsager critical spatial regularity for any 0 we show the existence of continuous periodic weak solutions v of the euler equations that do not conserve the kinetic energy and belong to the space lt1 cx1 3 e namely x v x t is e holder continuous in space at a e time t and the integral t 1 3 edt is finite a well known open conjecture of l onsager claims that such solutions exist even in the class lt cx1 3 e 2016 wiley periodicals inc

beltrami fields with a nonconstant proportionality factor are rare we consider the existence of beltrami fields with a nonconstant proportionality factor f in an open subset u of mathbb r 3 by reformulating this problem as a constrained evolution equation on a surface we find an explicit differential equation that f must satisfy whenever there is a nontrivial beltrami field with this factor this ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the c k topology k geqq 7 in particular there are no nontrivial beltrami fields whenever f has a regular level set diffeomorphic to the sphere this provides an explanation of the helical flow paradox of morgulis et al commun pure appl math 48 571 582 1995

on the energy of inviscid singular flows abstract it is known that the energy of a weak solution to the euler equation is conserved if it is slightly more regular than the besov space b 3 1 3 when the singular set of the solution is or belongs to a smooth manifold we derive various l p space regularity criteria dimensionally equivalent to the critical one in particular if the singular set is a hypersurface the energy of u is conserved provided the one sided non tangential limits to the surface exist and the non tangential maximal function is l 3 integrable while the maximal function of the pressure is l 3 2 integrable the results directly apply to prove energy conservation of the classical vortex sheets in both 2d and 3d at least in those cases where the energy is finite

onsager s conjecture almost everywhere in time in recent works by isett holder continuous euler flows in three dimensions with compact support in time pp 1 173 2012 and later by buckmaster et al ann math 2015 iterative schemes were presented for constructing solutions belonging to the holder class c 1 5 e of the 3d incompressible euler equations which do not conserve the total kinetic energy the cited work is partially motivated by a conjecture of lars onsager in 1949 relating to the existence of c 1 3 e solutions to the euler equations which dissipate energy in this note we show how the later scheme can be adapted in order to prove the existence of non trivial holder continuous solutions which for almost every time belong to the critical onsager holder regularity c 1 3 e and have compact temporal support

localised relative energy and finite speed of propagation for compressible flows abstract for the incompressible and the isentropic compressible euler equations in arbitrary space dimension we establish the principle of localised relative energy thus generalising the well known relative energy method to this end we adapt classical arguments of c dafermos to the euler equations we give several applications to the behaviour of weak solutions like local weak strong uniqueness local preservation of smoothness and finite speed of propagation for the isentropic system

self similar 2d euler solutions with mixed sign vorticity we construct a class of self similar 2d incompressible euler solutions that have initial vorticity of mixed sign the boundaries between regions of positive and negative vorticity form algebraic spirals similar to the kaden spiral and as opposed to prandtl s logarithmic vortex spirals also unlike the prandtl case spirals are not initially present

existence and uniqueness for stochastic 2d euler flows with bounded vorticity the strong existence and the pathwise uniqueness of solutions with l infty vorticity of the 2d stochastic euler equations are proved the noise is multiplicative and it involves the first derivatives a lagrangian approach is implemented where a stochastic flow solving a nonlinear flow equation is constructed the stability under regularizations is also proved

radial weak solutions for the perona malik equation as a differential inclusion abstract the perona malik equation is an ill posed forward backward parabolic equation with some application in image processing in this paper we study the perona malik type equation on a ball in an arbitrary dimension n and show that there exist infinitely many radial weak solutions to the homogeneous neumann boundary problem for smooth nonconstant radially symmetric initial data our approach is to reformulate the n dimensional equation into a one dimensional equation to convert the one dimensional problem into an inhomogeneous partial differential inclusion problem and to apply a baire s category method to the differential inclusion to generate infinitely many solutions

extensions of lieb s concavity theorem the operator function a b trf a b k k defined in pairs of bounded self adjoint operators in the domain of a function f of two real variables is convex for every hilbert schmidt operator k if and only if f is operator convex we obtain as a special case a new proof of lieb s concavity theorem for the function a b tra p k b q k where p and q are non negative numbers with sum p q 1 in addition we prove concavity of the operator function a b to tr left frac a a mu 1 k frac x1b b mu 2 k right in its natural domain d 2 1 2 cf definition 3

convex integration for a class of active scalar equations we show that a general class of active scalar equa tions including porous media and certain magnetostrophic tur bulence models admit non unique weak solutions in the class of bounded functions the proof is based upon the method of convex integration recently implemented for equations of fluid dynamics in 2 3

the energy balance relation for weak solutions of the density dependent navier stokes equations abstract we consider the incompressible inhomogeneous navier stokes equations with constant viscosity coefficient and density which is bounded and bounded away from zero we show that the energy balance relation for this system holds for weak solutions if the velocity density and pressure belong to a range of besov spaces of smoothness 1 3 a density dependent version of the classical karman howarth monin relation is derived

weak solutions with decreasing energy of incompressible euler equations weak solution of the euler equations is an l2 vector field u x t satisfying certain integral relations which express incompressibility and the momentum balance our conjecture is that some weak solutions are limits of solutions of viscous and compressible fluid equations as both viscosity and compressibility tend to zero thus we believe that weak solutions describe turbulent flows with very high reynolds numbers

on the nonuniqueness of weak solution of the euler equation weak solution of the euler equations is defined as an l2 vector field satisfying the integral relations expressing the mass and momentum balance their general nature has been quite unclear in this work an example of a weak solution on a 2 dimensional torus is constructed that is identically zero outside a finite time interval this example is simpler and more transparent than the previous example of v scheffer j geom anal 3 4 1993 pp 343 401 1997 john wiley sons inc

non uniqueness of weak solutions to 2d hypoviscous navier stokes equations abstract through an adaption of the convex integration scheme in the two dimensional case a result of the h principle type is presented for the two dimensional hypoviscous incompressible navier stokes equations it is shown that the c t 0 l x 2 weak solutions can possess compact temporal supports and thus are not unique in general

global existence of weak solutions to the three dimensional euler equations with helical symmetry abstract in this paper we mainly investigate the weak solutions of the three dimensional incompressible euler equations with helical symmetry in the whole space when the helical swirl vanishes specifically we establish the global existence of weak solutions when the initial vorticity lies in l 1 l p with p 1 our result extends the previous work 2 where the initial vorticity is compactly supported and belongs to l p with p 4 3 the key ingredient in this paper involves the explicit analysis of biot savart law with helical symmetry in domain r 2 via the theories of singular integral operators and second order elliptic equations

weak strong uniqueness for measure valued solutions we prove the weak strong uniqueness for measure valued solutions of the incompressible euler equations these were introduced by diperna and majda in their landmark paper commun math phys 108 4 667 689 1987 where in particular global existence to any l2 initial data was proven whether measure valued solutions agree with classical solutions if the latter exist has apparently remained open

on the energy dissipation rate of solutions to the compressible isentropic euler system in this paper we extend and complement the results in chiodaroli et al global ill posedness of the isentropic system of gas dynamics 2014 on the well posedness issue for weak solutions of the compressible isentropic euler system in 2 space dimensions with pressure law p 1 first we show that every riemann problem whose one dimensional self similar solution consists of two shocks admits also infinitely many two dimensional admissible bounded weak solutions not containing vacuum generated by the method of de lellis and szekelyhidi ann math 170 1417 1436 2009 arch ration mech anal 195 225 260 2010 moreover we prove that for some of these riemann problems and for 1 3 such solutions have a greater energy dissipation rate than the self similar solution emanating from the same riemann data we therefore show that the maximal dissipation criterion proposed by dafermos in j diff equ 14 202 212 1973 does not favour the classical self similar solutions

singular weak and absent solutions of the euler equations we will describe necessary and sufficient conditions for blowup and discuss weak solutions for the incompressible euler equations we will also describe a result concerning anomalous dissipation of energy

piecewise constant subsolutions for the muskat problem we show the existence of infinitely many admissible weak solutions for the incompressible porous media equations for all muskat type initial data with c 3 alpha regularity of the interface in the unstable regime and for all non horizontal data with c 3 alpha regularity in the stable regime our approach involves constructing admissible subsolutions with piecewise constant densities this allows us to give a rather short proof where it suffices to calculate the velocity and acceleration at time zero thus emphasizing the instantaneous nature of non uniqueness due to discontinuities in the initial data

h principles for the incompressible euler equations in dissipative euler flows and onsager s conjecture arxiv 1205 3626 preprint 2012 de lellis and szekelyhidi construct holder continuous dissipative weak solutions to the incompressible euler equations in the torus mathbb t 3 the construction consists of adding fast oscillations to the trivial solution we extend this result by establishing optimal h principles in two and three space dimensions specifically we identify all subsolutions defined in a suitable sense which can be approximated in the h 1 norm by exact solutions furthermore we prove that the flows thus constructed on mathbb t 3 are genuinely three dimensional and are not trivially obtained from solutions on mathbb t 2

directional oscillations concentrations and compensated compactness via microlocal compactness forms this work introduces microlocal compactness forms mcfs as a new tool to study oscillations and concentrations in l p bounded sequences of functions decisively mcfs retain information about the location value distribution and direction of oscillations and concentrations thus extending at the same time the theories of generalized young measures and h measures in l p spaces oscillations and concentrations precisely discriminate between weak and strong compactness and thus mcfs allow one to quantify the difference in compactness the definition of mcfs involves a fourier variable whereby differential constraints on the functions in the sequence can also be investigated easily a distinct advantage over young measure theory furthermore pointwise restrictions are reflected in the mcf as well paving the way for applications to tartar s framework of compensated compactness consequently we establish a new weak to strong compactness theorem in a geometric way after developing several aspects of the abstract theory we consider three applications for lamination microstructures the hierarchy of oscillations is reflected in the mcf the directional information retained in an mcf is harnessed in the relaxation theory for anisotropic integral functionals finally we indicate how the theory pertains to the study of propagation of singularities in certain systems of pdes the proofs combine measure theory young measures and harmonic analysis

the cubic to orthorhombic phase transition rigidity and non rigidity properties in the linear theory of elasticity in this paper we investigate the cubic to orthorhombic phase transition in the framework of linear elasticity using convex integration techniques we prove that this phase transition represents one of the simplest three dimensional examples in which already the linearised theory of elasticity displays non rigidity properties as a complementary result we demonstrate that surface energy constraints rule out such highly oscillatory behaviour we give a full characterization of all possibly emerging patterns for generic material parameters

optimal existence theorems for nonhomogeneous differential inclusions abstract in this paper we address the question of solvability of the differential inclusions 1 1 our approach to these problems is based on the idea of constructing a sequence of approximate solutions which converges strongly and makes use of gromov s idea following earlier work of nash and kuiper to control convergence of the gradients by appropriate selection of the elements of the sequence in this paper we identify an optimal setting of this method in particular we show that the existence result holds for general upper semicontinuous functions h without extra requirements like quasiconvexity of h with respect to du which was assumed in previous works where the idea to apply the baire category lemma to the sets of approximate solutions was developed we also apply our result to identify the minimal sets where the function h should vanish to guarantee solvability of the inclusions

maps on positive definite matrices preserving bregman and jensen divergences abstract in this paper we determine those bijective maps of the set of all positive definite n n complex matrices which preserve a given bregman divergence corresponding to a differentiable convex function that satisfies certain conditions we cover the cases of the most important bregman divergences and present the precise structure of the mentioned transformations similar results concerning jensen divergences and their preservers are also given

euler and navier stokes equations on the hyperbolic plane we show that nonuniqueness of the leray hopf solutions of the navier stokes equation on the hyperbolic plane 2 observed by chan and czubak is a consequence of the hodge decomposition we show that this phenomenon does not occur on n whenever n 3 we also describe the corresponding general hamiltonian framework of hydrodynamics on complete riemannian manifolds which includes the hyperbolic setting

non uniqueness for an energy critical heat equation on mathbb r 2 r2 we construct a singular solution of a stationary nonlinear schrodinger equation on mathbb r 2 with square exponential nonlinearity having linear behavior around zero in view of trudinger moser inequality this type of nonlinearity has an energy critical growth we use this singular solution to prove non uniqueness of mild solutions for the cauchy problem of the corresponding semilinear heat equation the proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space

a sufficient condition for the kolmogorov 4 5 law for stationary martingale solutions to the 3d navier stokes equations we prove that statistically stationary martingale solutions of the 3d navier stokes equations on mathbb t 3 subjected to white in time colored in space forcing satisfy the kolmogorov 4 5 law in an averaged sense and over a suitable inertial range using only the assumption that the kinetic energy is o nu 1 as nu rightarrow 0 where is the inverse reynolds number this plays the role of a weak anomalous dissipation no energy balance or additional regularity is assumed aside from that satisfied by all martingale solutions from the energy inequality if the force is statistically homogeneous then any homogeneous martingale solution satisfies the spherically averaged 4 5 law pointwise in space an additional hypothesis of approximate isotropy in the inertial range gives the traditional version of the kolmogorov law we demonstrate a necessary condition by proving that energy balance and an additional quantitative regularity estimate as nu rightarrow 0 imply that the 4 5 law or any similar scaling law cannot hold

existence and non uniqueness of global weak solutions to inviscid primitive and boussinesq equations we consider the initial value problem for the inviscid primitive and boussinesq equations in three spatial dimensions we recast both systems as an abstract euler type system and apply the methods of convex integration of de lellis and szekelyhidi to show the existence of infinitely many global weak solutions of the studied equations for general initial data we also introduce an appropriate notion of dissipative solutions and show the existence of suitable initial data which generate infinitely many dissipative solutions

young measures generated by ideal incompressible fluid flows in their seminal paper diperna and majda commun math phys 108 4 667 689 1987 introduced the notion of a measure valued solution for the incompressible euler equations in order to capture complex phenomena present in limits of approximate solutions such as persistence of oscillation and development of concentrations furthermore they gave several explicit examples exhibiting such phenomena in this paper we show that any measure valued solution can be generated by a sequence of exact weak solutions in particular this gives rise to a very large arguably too large set of weak solutions of the incompressible euler equations

on self similar solutions to the incompressible euler equations abstract recent numerical simulations have shown the existence of multiple self similar solutions to the cauchy problem for the 2 dimensional incompressible euler equation with initial vorticity in l l o c p r 2 1 p toward a rigorous validation of these computations in this paper we analytically construct self similar solutions i on an outer domain of the form x r and ii in a neighborhood of the points where the solution exhibits a spiraling vortex singularity the outer solution is obtained as the fixed point of a contractive transformation based on the biot savart formula and integration along characteristics the inner solution is constructed using a system of adapted coordinates following the approach of v elling 2016 17

statistical solutions of hyperbolic conservation laws foundations we seek to define statistical solutions of hyperbolic systems of conservation laws as time parametrized probability measures on p integrable functions to do so we prove the equivalence between probability measures on l p spaces and infinite families of correlation measures each member of this family termed a correlation marginal is a young measure on a finite dimensional tensor product domain and provides information about multi point correlations of the underlying integrable functions we also prove that any probability measure on a l p space is uniquely determined by certain moments correlation functions of the equivalent correlation measure we utilize this equivalence to define statistical solutions of multi dimensional conservation laws in terms of an infinite set of equations each evolving a moment of the correlation marginal these evolution equations can be interpreted as augmenting entropy measure valued solutions with additional information about the evolution of all possible multi point correlation functions our concept of statistical solutions can accommodate uncertain initial data as well as possibly non atomic solutions even for atomic initial data for multi dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting entropy statistical solutions exist are unique and are stable with respect to the 1 wasserstein metric on probability measures on l 1

non uniqueness and h principle for holder continuous weak solutions of the euler equations in this paper we address the cauchy problem for the incompressible euler equations in the periodic setting we prove that the set of holder 1 5 varepsilon wild initial data is dense in l 2 where we call an initial datum wild if it admits infinitely many admissible holder 1 5 varepsilon weak solutions we also introduce a new set of stationary flows which we use as a perturbation profile instead of beltrami flows in order to show that a general form of the h principle applies to holder continuous weak solutions of the euler equations our result indicates that in a deterministic theory of three dimensional turbulence the reynolds stress tensor can be arbitrary and need not satisfy any additional closure relation

energy conservation in two dimensional incompressible ideal fluids this note addresses the issue of energy conservation for the 2d euler system with an lp control on vorticity we provide a direct argument based on a mollification in physical space to show that the energy of a weak solution is conserved if omega nabla times u in l frac 3 2 an example of a 2d field in the class omega in l frac 3 2 epsilon for any epsilon 0 and u in b 1 3 3 infty onsager critical space see shvydkoy in discr contin dyn syst ser s 3 3 473 496 2010 is constructed with non vanishing energy flux this demonstrates sharpness of the kinematic argument which does not differentiate between 2d and 3d and requires onsager s regularity control on the solution next we show that for physically realizable solutions there is a mechanism preventing the anomalous dissipation in 2d that does not require such a control namely we prove that any solution to the euler equations produced via a vanishing viscosity limit from the navier stokes equations with omega in l p for p 1 conserves energy

cauchy problem for dissipative holder solutions to the incompressible euler equations we consider solutions to the cauchy problem for the incompressible euler equations on the 3 dimensional torus which are continuous or holder continuous for any exponent theta frac 1 16 using the techniques introduced in de lellis and szekelyhidi inventiones mathematicae 9 377 407 2013 dissipative euler flows and onsager s conjecture 2012 we prove the existence of infinitely many holder continuous initial vector fields starting from which there exist infinitely many holder continuous solutions with preassigned total kinetic energy

fluids elasticity geometry and the existence of wrinkled solutions we are concerned with underlying connections between fluids elasticity isometric embedding of riemannian manifolds and the existence of wrinkled solutions of the associated nonlinear partial differential equations in this paper we develop such connections for the case of two spatial dimensions and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the gauss codazzi equations through examples for the euler equations for fluids and the euler lagrange equations for elastic solids these results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics

attainment results for nematic elastomers we consider a class of non quasiconvex frame indifferent energy densities which includes ogden type energy densities for nematic elastomers for the corresponding geomet rically linear problem we provide an explicit minimizer of the energy functional satisfying a nontrivial boundary condition other attainment results both for the nonlinear and the lin earized model are obtained by using the theory of convex integration introduced by muller and u sverak in the context of crystalline solids

lack of uniqueness for weak solutions of the incompressible porous media equation in this work we consider weak solutions of the incompressible two dimensional porous media ipm equation by using the approach of de lellis szekelyhidi we prove non uniqueness for solutions in l in space and time

extremal solutions of differential inclusions via baire category a dual approach abstract let f be a continuous multifunction on r n with compact convex values for any vector w let f w x f x be the subset of points which maximize the inner product with w call w the set of all continuous functions w 0 t r n with the following property all solutions to the cauchy problem x t f w t x t x 0 0 are also solutions to x t ext f x t we prove that w is residual in c 0 t r n

on the weak solutions to the maxwell landau lifshitz equations and to the hall magneto hydrodynamic equations in this paper we deal with weak solutions to the maxwell landau lifshitz equations and to the hall magneto hydrodynamic equations first we prove that these solutions satisfy some weak strong uniqueness property then we investigate the validity of energy identities in particular we give a sufficient condition on the regularity of weak solutions to rule out anomalous dissipation in the case of the hall magneto hydrodynamic equations we also give a sufficient condition to guarantee the magneto helicity identity our conditions correspond to the same heuristic scaling as the one introduced by onsager in hydrodynamic theory finally we examine the sign locally of the anomalous dissipations of weak solutions obtained by some natural approximation processes

anomalous dissipation for 1 5 holder euler flows recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3d incompressible euler equations in holder spaces the motivation comes from onsager s conjecture the construction involves a superposition of weakly interacting perturbed beltrami flows on infinitely many scales an obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field in a recent paper the third author has improved upon the methods introducing some novel ideas on how to deal with this obstruction thereby reaching a better holder exponent albeit weaker than the one conjectured by onsager in this paper we give a shorter proof of this final result adhering more to the original scheme of the second and fourth authors and introducing some new devices more precisely we show that for any positive e there exist periodic solutions of the 3d incompressible euler equations that dissipate the total kinetic energy and belong to the holder class c 1 5 epsilon

the euler equations as a differential inclusion we propose a new point of view on weak solutions of the euler equations describing the motion of an ideal incompressible fluid in r n with n 2 we give a reformulation of the euler equations as a differential inclusion and in this way we obtain transparent proofs of several celebrated results of v scheffer and a shnirelman concerning the non uniqueness of weak solutions and the existence of energy decreasing solutions our results are stronger because they work in any dimension and yield bounded velocity and pressure