Convex Integration Research Articles

The Initial Value Problem for the Euler Equations of Incompressible Fluids Viewed as a Concave Maximization Problem

We consider the Euler equations of incompressible fluids and attempt to solve the initial value problem with the help of a concave maximization problem.We show that this problem, which shares a similar structure with the optimal transport problemwith quadratic cost, in its "Benamou-Brenier" formulation,always admits a relaxed solution that can be interpretedin terms of $sub-solution$ of the Euler equations in the sense of convex integration theory.Moreover, any smooth solution of the Euler equations can be recovered from this maximization problem, at least for short times.

On one-dimensional forward–backward diffusion equations with linear convection and reaction

We study the initial-Neumann boundary value problem for a class of one-dimensional forward–backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to contain two forward-diffusion phases. We prove that for all smooth initial data with derivative value lying in certain phase transition regions one can construct infinitely many Lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases. Furthermore, we introduce a notion of transition gauge for such solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant. The results are new even for the pure forward–backward diffusion problem without convection and reaction. Our primary approach relies on a combination of the convex integration and Baire's category methods for a related nonlocal differential inclusion.

Ill-Posedness of Leray Solutions for the Hypodissipative Navier–Stokes Equations

We prove the ill-posedness of Leray solutions to the Cauchy problem for the ipodissipative Navier--Stokes equations, when the dissipative term is a fractional Laplacian $(-\Delta)^\alpha$ with exponent $\alpha < \frac{1}{5}$. The proof follows the ''convex integration methods'' introduced by the second author and L\'aszl\'o Sz\'ekelyhidi Jr. for the incomprresible Euler equations. The methods yield indeed some conclusions even for exponents in the range $[\frac{1}{5}, \frac{1}{2}[$.

On asymptotic behavior and energy distribution for some one-dimensional non-parabolic diffusion problems

We study some non-parabolic diffusion problems in one space dimension, where the diffusion flux exhibits forward and backward nature of the Perona–Malik, Höllig or non-Fourier type. Classical weak solutions to such problems are constructed in a way to capture some expected and unexpected properties, including anomalous asymptotic behaviors and energy dissipation or allocation. Specific properties of solutions will depend on the type of the diffusion flux, but the primary method of our study relies on reformulating diffusion equations involved as an inhomogeneous partial differential inclusion and on constructing solutions from the differential inclusion by a combination of the convex integration and Baire’s category methods. In doing so, we introduce the appropriate notion of subsolutions of the partial differential inclusion and their transition gauge, which plays a pivotal role in dealing with some specific features of the constructed weak solutions.

Refinements of the holonomic approximation lemma

The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the $h-$principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working one pure partial derivative at a time is central to the discussion. We give applications of our results to flexible symplectic and contact topology.

Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Dirichlet Problem with Affine Data in int(K^{lc})

In this article we continue our study of higher Sobolev regularity of flexible convex integration solutions to differential inclusions arising from applications in materials sciences. We present a ...

Convex Integration for Scalar Conservation Laws in One Space Dimension

In the absence of the entropy condition, we construct an $L^\infty$ solution to the Cauchy problem of a scalar conservation law in one space dimension that exhibits fine-scale oscillations in the i...

$$T_5$$ T 5 -Configurations and non-rigid sets of matrices

In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation of the rank-one convex hull for this specific set was later carried out in 2010 by W. Pompe in Calc. Var. PDE 37(3–4):461–473, (2010) by delicate geometric arguments. For more general sets of matrices a full computation of the rank-one convex hull is clearly out of reach. Therefore, in this short note we revisit the construction and propose a new, in some sense generic method for deciding whether convex integration for a given set of matrices can be carried out, which does not require the full computation of the rank-one convex hull.

The Riemann Problem for the Multidimensional Isentropic System of Gas Dynamics is Ill-Posed if It Contains a Shock

In this paper we consider the isentropic compressible Euler equations in two space dimensions together with particular initial data. The latter consists only of two constant states, where one state lies on the lower and the other state on the upper half plane. The aim is to investigate if there exists a unique entropy solution or if the convex integration method produces infinitely many entropy solutions. For some initial states this question has been answered by E. Feireisl and O. Kreml, and also G.-Q. Chen and J. Chen, where there exists a unique entropy solution. For other initial states E. Chiodaroli, O. Kreml and C. De Lellis showed that there are infinitely many entropy solutions. For still other initial states the question on uniqueness remained open and this will be the content of this paper. This paper can be seen as a completion of the aforementioned papers by showing that the solution is non-unique in all cases (except if the solution is smooth).

Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions

We study the higher gradient integrability of distributional solutions u to the equation {{mathrm{div}}}(sigma nabla u) = 0 in dimension two, in the case when the essential range of sigma consists of only two elliptic matrices, i.e., sigma in {sigma _1, sigma _2} a.e. in Omega . In Nesi et al. (Ann Inst H Poincaré Anal Non Linéaire 31(3):615–638, 2014), for every pair of elliptic matrices sigma _1 and sigma _2, exponents p_{sigma _1,sigma _2}in (2,+infty ) and q_{sigma _1,sigma _2}in (1,2) have been found so that if uin W^{1,q_{sigma _1,sigma _2}}(Omega ) is solution to the elliptic equation then nabla uin L^{p_{sigma _1,sigma _2}}_{mathrm{weak}}(Omega ) and the optimality of the upper exponent p_{sigma _1,sigma _2} has been proved. In this paper we complement the above result by proving the optimality of the lower exponent q_{sigma _1,sigma _2}. Precisely, we show that for every arbitrarily small delta , one can find a particular microgeometry, i.e., an arrangement of the sets sigma ^{-1}(sigma _1) and sigma ^{-1}(sigma _2), for which there exists a solution u to the corresponding elliptic equation such that nabla u in L^{q_{sigma _1,sigma _2}-delta }, but nabla u notin L^{q_{sigma _1,sigma _2}}. The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in Astala et al. (Ann Scuola Norm Sup Pisa Cl Sci 5(7):1–50, 2008) for the isotropic case.

Convex integration for the Monge–Ampère equation in two dimensions

This paper concerns the questions of flexibility and rigidity of solutions to the Monge-Amp\`ere equation which arises as a natural geometrical constraint in prestrained nonlinear elasticity. In particular, we focus on anomalous i.e. "flexible" weak solutions that can be constructed through methods of convex integration \`a la Nash & Kuiper and establish the related h-principle for the Monge-Amp\`ere equation in two dimensions

Failure of the chain rule for the divergence of bounded vector fields

We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. For instance, we can construct defects which are absolutely continuous with respect to the Lebesgue measure, or defects which are not even measures.

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 Sz\'ekelyhidi 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.

Weak solutions for Euler systems with non-local interactions

We consider several modifications of the Euler system of fluid dynamics, including its pressureless variant driven by non-local interaction repulsive–attractive and alignment forces in the space dimension N = 2 , 3 . These models arise in the study of self-organization in collective behavior modeling of animals and crowds. We adapt the method of convex integration to show the existence of infinitely many global-in-time weak solutions for any bounded initial data. Then we consider the class of dissipative solutions satisfying, in addition, the associated global energy balance (inequality). We identify a large set of initial data for which the problem admits infinitely many dissipative weak solutions. Finally, we establish a weak–strong uniqueness principle for the pressure-driven Euler system with non-local interaction terms as well as for the pressureless system with Newtonian interaction.

Bi-Sobolev homeomorphisms f with Df and $$Df^{-1}$$ D f - 1 of low rank using laminates

Let \(\Omega \subset \mathbb {R}^{n}\) be a bounded open set. Given \(1\le m_1,m_2\le n-2\), we construct a homeomorphism \(f :\Omega \rightarrow \Omega \) that is Holder continuous, f is the identity on \(\partial \Omega \), the derivative Df has rank \(m_1\) a.e. in \(\Omega \), the derivative \(D f^{-1}\) of the inverse has rank \(m_2\) a.e. in \(\Omega \), \(Df\in W^{1,p}\) and \(Df^{-1}\in W^{1,q}\) for \(p<\min \{m_1+1,n-m_2\}\), \(q<\min \{m_2+1,n-m_1\}\). The proof is based on convex integration and laminates. We also show that the integrability of the function and the inverse is sharp.

Sobolev homeomorphisms with gradients of low rank via laminates

Abstract Let Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} be a bounded open set. Given 2 ≤ m ≤ n {2\leq m\leq n} , we construct a convex function u : Ω → ℝ {u\colon\Omega\to\mathbb{R}} whose gradient f = ∇ ⁡ u {f=\nabla u} is a Hölder continuous homeomorphism, f is the identity on ∂ ⁡ Ω {\partial\Omega} , the derivative Df has rank m - 1 {m-1} a.e. in Ω and Df is in the weak L m {L^{m}} space L m , w {L^{m,w}} . The proof is based on convex integration and staircase laminates.

ORIENTATION-PRESERVING YOUNG MEASURES

We prove a characterization result in the spirit of the Kinderlehrer–Pedregal Theorem for Young measures generated by gradients of Sobolev maps satisfying the orientation-preserving constraint, that is, the pointwise Jacobian is positive almost everywhere. The argument to construct the appropriate generating sequences from such Young measures is based on a variant of convex integration in conjunction with an explicit lamination construction in matrix space. Our generating sequence is bounded in L^p for p less than the space dimension, a regime in which the pointwise Jacobian behaves flexibly, as is illustrated by our results. On the other hand, for p larger than or equal to the space dimension the situation necessarily becomes rigid and a construction as presented here cannot succeed. Applications to relaxation of integral functionals, the theory of semiconvex hulls and approximation of weakly orientation-preserving maps by strictly orientation-preserving ones in Sobolev spaces are given.

An overview of some recent results on the Euler system of isentropic gas dynamics

This overview is concerned with the well-posedness problem for the isentropic compressible Euler equations of gas dynamics. The results we present are in line with the programof investigatingthe efficiency of different selection criteria proposed in the literature in order to weed out non-physical solutions to more-dimensional systems of conservation laws and they build upon the method of convex integration developed by De Lellis and Szekelyhidi for the incompressible Euler equations. Mainly following [5], we investigate the role of the maximal dissipation criterion proposed by Dafermos in [6]: we prove how, for specific pressure laws, some non-standard (i.e. constructed via convex integration methods) solutions to the Riemann problem for the isentropic Euler system in two space dimensions have greater energy dissipation rate than the classical self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos does not favour in general the self-similar solutions.

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.

Non-uniqueness of admissible weak solutions to compressible Euler systems with source terms

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 Székelyhidi [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.