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Baisheng Yan

Publications and source records attributed to Baisheng Yan.

17 recordsLinked to original sources
arXiv2026

Divergence-form equations admitting nowhere $C^1$ Lipschitz weak solutions

We study a class of partial differential equations in divergence form that admit highly irregular Lipschitz weak solutions. By reformulating these divergence-form equations as a first-order partial differential relation and adapting the convex integration scheme recently developed in \cite{GKY26} for irregular diffusion equations, we show that the same structural Condition~$O_N$ introduced there also ensures the existence of Lipschitz weak solutions that are nowhere $C^1$ for the corresponding time-independent equations in bounded domains, under suitable boundary data. In particular, for the smooth strongly polyconvex functions on $\mathbb{R}^{2\times n}$ constructed in that paper for all $n \ge 2$, the associated Euler--Lagrange equations admit Lipschitz weak solutions that are nowhere $C^1$ and satisfy zero boundary conditions in any bounded domain of $\mathbb{R}^n$. Our approach relies on new building blocks constructed from the same wave cone and $\mathcal{T}_N$-configurations employed in the analysis of diffusion equations.

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arXiv2026

Irregular Diffusions and Loss of Regularity in Polyconvex Gradient Flows

We investigate diffusion-type partial differential equations that are irregular in the sense that they admit weak solutions which are nowhere smooth, even for prescribed smooth data. By reformulating these equations as first-order partial differential relations and adapting the method of convex integration, we develop a construction scheme based on new geometric structures, referred to as $\mathcal{T}_N$-configurations, together with a simplified structural hypothesis on the diffusion functions, termed Condition $O_N$. Under this condition, we show that the associated initial and boundary value problems with certain smooth initial-boundary data admit infinitely many Lipschitz weak solutions that are nowhere $C^1$. We further analyze specific $\mathcal{T}_N$-configurations and establish nondegeneracy conditions that are essential for verifying Condition $O_N$. As an application, we construct examples of strongly polyconvex energy functionals whose gradient flows generate irregular diffusion equations, thereby revealing a failure of regularity and uniqueness even within the class of polyconvex gradient flows.

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arXiv2023

On integral convexity, variational solutions and nonlinear semigroups

In this paper we provide a different approach for existence of the variational solutions of the gradient flows associated to functionals on Sobolev spaces studied in \cite{BDDMS20}. The crucial condition is the convexity of the functional under which we show that the variational solutions coincide with the solutions generated by the nonlinear semigroup associated to the functional. For integral functionals of the form $\mathbf F(u)=\int_Ωf(x,Du(x)) dx,$ where $f(x,ξ)$ is $C^1$ in $ξ$, we also make some remarks on the connections between convexity of $\mathbf F$ (called the integral convexity of $f$) and certain monotonicity conditions of the gradient map $D_ξf.$ In particular, we provide an example to show that even for functions of the simple form $f=f(ξ)$, the usual quasimonotonicity of $D_ξf$ is not sufficient for the integral convexity of $f.$

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arXiv2023

On higher integrability for $p(x)$-Laplacian equations with drift

In this paper, we study the higher integrability for the gradient of weak solutions of $p(x)$-Laplacians equation with drift terms. We prove a version of generalized Gehring's lemma under some weaker condition on the modulus of continuity of variable exponent $p(x)$ and present a modified version of Sobolev-Poincaré inequality with such an exponent. When $p(x)>2$ we derive the reverse Hölder inequality with a proper dependence on the drift and force terms and establish a specific high integrability result. Our condition on the exponent $p(x)$ is more specific and weaker than the known conditions and our results extend some results on the $p(x)$-Laplacian equations without drift terms.

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arXiv2023

The $τ_N$-configurations and polyconvex gradient flows

We study a generalization of $T_N$-configurations, called the $τ_N$-configurations, for constructing certain irregular solutions of some nonlinear diffusion systems by the method of convex integration. We construct some polyconvex functions that support a parametrized family of $τ_N$-configurations satisfying a general openness condition; this will guarantee the existence of nowhere-$C^1$ Lipschitz weak solutions to the initial boundary value problems of the polyconvex gradient flows. We elaborate on such constructions and the subsequent verification of the openness condition when the dimension is at least 4 to avoid some complicated calculations that cannot be done by hand but would otherwise be needed for dimensions 2 and 3.

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arXiv2022

On a class of special Euler-Lagrange equations

We make some remarks on the Euler-Lagrange equation of energy functional $I(u)=\int_Ωf(\det Du)\,dx,$ where $f\in C^1(\mathbb R).$ For certain weak solutions $u$ we show that the function $f'(\det Du)$ must be a constant over the domain $Ω$ and thus, when $f$ is convex, all such solutions are an energy minimizer of $I(u).$ However, other weak solutions exist such that $f'(\det Du)$ is not constant on $Ω.$ We also prove some results concerning the homeomorphism solutions, non-quasimonotonicty, radial solutions, and some special properties and questions in the 2-D cases.

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arXiv2022

On nonuniqueness and nonregularity for gradient flows of polyconvex functionals

We provide some counterexamples concerning the uniqueness and regularity of weak solutions to the initial-boundary value problem for gradient flows of certain strongly polyconvex functionals by showing that such a problem can possess a trivial classical solution as well as infinitely many weak solutions that are nowhere smooth. Such polyconvex functions have been constructed in the previous work, and the nonuniqueness and nonregularity will be achieved by reformulating the gradient flow as a space-time partial differential relation and then using the convex integration method to construct certain strongly convergent sequences of subsolutions that have a uniform control on the local essential oscillations of their spatial gradients.

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arXiv2019

Convex Integration for Diffusion Equations

We study the initial-boundary value problem for a class of diffusion equations with nonmonotone diffusion flux functions, including forward-backward parabolic equations and the gradient flows of nonconvex energy functionals, under the framework of partial differential inclusions using the method of convex integration and Baire's category. In connection with rank-one convex hulls of the corresponding matrix sets, we introduce a structural condition on the diffusion flux function, called Condition (OC), and establish the nonuniqueness and instability of Lipschitz solutions to the initial-boundary value problem under this condition.

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arXiv2019

Lipschitz Solutions for the Gradient Flow of Polyconvex Functionals

In this sequel to a previous paper, we construct certain smooth strongly polyconvex functions $F$ on $\mathbb M^{2\times 2}$ such that $σ=DF$ satisfies the Condition (OC) in that paper. As a result, we show that the initial-boundary value problem for the gradient flow of such polyconvex energy functionals is highly ill-posed even for some smooth initial-boundary data in the sense that the problem possesses a weakly* convergent sequence of Lipschitz weak solutions whose limit is not a weak solution.

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arXiv2018

On weak closure of some diffusion equations

We study the closure of approximating sequences of some diffusion equations under certain weak convergence. A specific description of the closure under weak $H^1$-convergence is given, which reduces to the original equation when the equation is parabolic. However, the closure under strong $L^2$-convergence may be much larger, even for parabolic equations.

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arXiv2016

Non-parabolic diffusion problems in one space dimension

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.

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arXiv2016

Two-phase forward solutions for one-dimensional forward-backward parabolic equations with linear convection and reaction

We study the existence and properties of Lipschitz continuous weak solutions to the Neumann boundary value problem for a class of one-dimensional quasilinear forward-backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to be of a forward-backward type that contains two forward-diffusion phases. We prove that, for all smooth initial data, there exists at least one weak solution whose spatial derivative stays in the two forward phases. Also, for all smooth initial data that have a derivative value lying in a certain phase transition range, we show that there exist infinitely many such solutions that exhibit instantaneous phase transitions between the two forward phases. Moreover, we introduce the notion of transition gauge for such forward solutions and prove that the gauge of all constructed two-phase forward solutions can be arbitrarily close to a certain fixed constant.

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arXiv2015

On Lipschitz solutions for some forward-backward parabolic equations

We investigate the existence and properties of Lipschitz solutions for some forward-backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Finally, we study two important cases of forward-backward anisotropic diffusion in which the density condition can be realized and therefore the existence results follow together with micro-oscillatory behavior of solutions. The first case is a generalization of the Perona-Malik model in image processing and the other that of Höllig's model related to the Clausius-Duhem inequality in the second law of thermodynamics.

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arXiv2015

Convex integration and infinitely many weak solutions to the Perona-Malik equation in all dimensions

We prove that for all smooth nonconstant initial data the initial-Neumann boundary value problem for the Perona-Malik equation in image processing possesses infinitely many Lipschitz weak solutions on smooth bounded convex domains in all dimensions. Such existence results have not been known except for the one-dimensional problems. Our approach is motivated by reformulating the Perona-Malik equation as a nonhomogeneous partial differential inclusion with linear constraint and uncontrollable components of gradient. We establish a general existence result by a suitable Baire's category method under a pivotal density hypothesis. We finally fulfill this density hypothesis by convex integration based on certain approximations from an explicit formula of lamination convex hull of some matrix set involved.

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arXiv2013

A Landau de Gennes theory of liquid crystal elastomers

In this article, we study minimization of the Landau-de Gennes energy for liquid crystal elastomer.The total energy, is of the sum of the Lagrangian elastic stored energy function of the elastomer and the Eulerian Landau-de Gennes energy of the liquid crystal. Our model consider two sources of anisotropy represented by the traceless nematic order tensor $Q$ (rigid units), and the positive definite step-length tensor $L$ (network). This work is motivated by the study of cytoskeletal networks which can be regarded as consisting of rigid rod units crosslinked into a polymeric-type network. Due to the mixed Eulerian-Lagrangian structure of the energy, it is essential that the deformation maps $φ$ be invertible. We require sufficient regularity of the fields $(φ, Q)$ of the problem, and that the deformation map satisfies the Ciarlet-Necas condition. These, in turn, determine what boundary conditions are admissible, which include the case of Dirichlet conditions on both fields. The approach of including the Rapini-Papoular surface energy for the pull-back tensor $\tilde Q$ is also discussed. The regularity requirements lead us to consider powers of the gradient of the order tensor $Q$ higher than quadratic in the energy. We assume polyconvexity of the stored energy function with respect to the effective deformation tensor and apply methods from isotropic nonlinear elasticity. We formulate a necessary and sufficient condition to guarantee this invertibility property in terms of the growth to infinity of the bulk liquid crystal energy $f(Q)$, as the minimum eigenvalue of $Q$ approaches the singular limit of $-\frac{1}{3}$. $L$ becomes singular as the minimum eigenvalue of $Q$ reaches $-\frac{1}{3}$. Lower bounds on the eigenvalues of $Q$ are needed to ensure compatibility between the theories of Landau-de Gennes and Maier-Saupe of nematics (see Ball 2010).

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arXiv2013

Radial weak solutions for the Perona-Malik equation as a differential inclusion

The Perona-Malik equation is an ill-posed forward-backward parabolic equation with major application in image processing. In this paper we study the Perona-Malik type equation and show that, in all dimensions, there exist infinitely many radial weak solutions to the homogeneous Neumann boundary problem for any 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 a differential inclusion problem, and to apply a Baire's category method to generate infinitely many solutions.

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