This paper is concerned with the development and analysis of a mathematical model that is motivated by interstitial hydrodynamics and tissue deformation mechanics (poro-elasto-hydrodynamics) within an in-vitro solid tumour. The classical mixture theory is adopted for mass and momentum balance equations for a two-phase system. A main contribution of this study is we treat the physiological transport parameter (i.e., hydraulic resistivity) as anisotropic and heterogeneous, thus the governing system is strongly coupled and non-linear. We derived a weak formulation and then formulated the equivalent fixed-point problem. This enabled us to use the Galerkin method, and the classical results on monotone operators combined with the well-known Schauder and Banach fixed-point theorems to prove the existence and uniqueness of results.

This work is devoted to the study of the sub-critical case of an anisotropic fully parabolic Keller–Segel chemotaxis system. We prove the existence of nonnegative weak solutions of (1.1) without restriction on the size of the initial data.

Structural changes of the pore space and clogging phenomena are inherent to many porous media applications. However, related analytical investigations remain challenging due to potentially vanishing coefficients in the respective systems of partial differential equations. In this research, we apply an appropriate scaling of the unknowns and work with porosity-weighted function spaces. This enables us to prove existence, uniqueness and non-negativity of weak solutions to a combined flow and transport problem with vanishing, but prescribed porosity field, permeability and diffusion.

This paper concerns the energy conservation for the weak solutions of the compressible Navier–Stokes equations. Assume that the density is positively bounded, we work on the regularity assumption on the gradient of the velocity, and establish a Lp–Ls type condition for the energy equality to hold in the distributional sense in time. We mention that no regularity assumption on the density derivative is needed any more.

We study the existence of entropy solutions by assuming the right-hand side function f to be an integrable function for some elliptic nonlocal p-Laplacian type problems. Moreover, the existence of weak solutions for the corresponding parabolic cases is also established. The main aim of this paper is to provide some positive answers for the two questions proposed by Chipot and de Oliveira (Math. Ann., 2019, 375, 283-306).

We give a weak-Lp Serrin-type regularity criterion for a weak solution to the three-dimensional magnetohydrodynamics equations in a bounded domain Ω ⊂ ℝ3.

We prove the existence of weak solutions of complex $m$ -Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$ , $p>n/m$ ( $n$  is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

We consider a diffuse interface model for tumour growth consisting of a Cahn–Hilliard equation with source terms coupled to a reaction–diffusion equation. The coupled system of partial differential equations models a tumour growing in the presence of a nutrient species and surrounded by healthy tissue. The model also takes into account transport mechanisms such as chemotaxis and active transport. We establish well-posedness results for the tumour model and a variant with a quasi-static nutrient. It will turn out that the presence of the source terms in the Cahn–Hilliard equation leads to new difficulties when one aims to derive a priori estimates. However, we are able to prove continuous dependence on initial and boundary data for the chemical potential and for the order parameter in strong norms.

We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruud et al. ((2013) J. Math. Biol.67 1457–1485). This model consists of the Cahn–Hilliard equation for the tumour cell fraction ϕ nonlinearly coupled with a reaction–diffusion equation for ψ, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function p(ϕ) multiplied by the differences of the chemical potentials for ϕ and ψ. The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of ϕ + ψ. Here, we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.

In this paper the existence of infinitely many solutions for a class of Kirchhoff-type problems involving the p-Laplacian, with p > 1, is established. By using variational methods, we determine unbounded real intervals of parameters such that the problems treated admit either an unbounded sequence of weak solutions, provided that the nonlinearity has a suitable behaviour at ∞, or a pairwise distinct sequence of weak solutions that strongly converges to 0 if a similar behaviour occurs at 0. Some comparisons with several results in the literature are pointed out. The last part of the work is devoted to the autonomous elliptic Dirichlet problem.

The purpose of this paper is to apply particle methods to the numerical solution of theEPDiff equation. The weak solutions of EPDiff are contact discontinuities that carrymomentum so that wavefront interactions represent collisions in which momentum isexchanged. This behavior allows for the description of many rich physical applications,but also introduces difficult numerical challenges. We present a particle method for theEPDiff equation that is well-suited for this class of solutions and for simulatingcollisions between wavefronts. Discretization by means of the particle method is shown topreserve the basic Hamiltonian, the weak and variational structure of the originalproblem, and to respect the conservation laws associated with symmetry under the Euclideangroup. Numerical results illustrate that the particle method has superior features in bothone and two dimensions, and can also be effectively implemented when the initial data ofinterest lies on a submanifold.

A class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

In this paper we consider the initialboundary value problem of a parabolic-elliptic system for imageinpainting, and establish the existence and uniqueness of weaksolutions to the system in dimension two.

Let $A = (a^{ij})$ be a Borel mapping on $[0, 1] \times \mathbb{R}^d$ with values in the space of non-negative operators on $\mathbb{R}^d$ and let $b = (b^i)$ be a Borel mapping on $[0, 1] \times \mathbb{R}^d$ with values in $\mathbb{R}^d$. Let

\[Lu(t, x) = \partial_{t} u(t, x) + a^{ij}(t, x)\partial_{x_i}\partial_{x_j}u(t, x) + b^i(t, x)\partial_{x_i}u(t, x), \quad u \in C_0^\infty((0, 1)\times \mathbb{R}^d).\]

Under broad assumptions on $A$ and $b$, we construct a family $\mu = (\mu_t)_{t \in [0, 1)}$ of probability measures $\mu_t$ on $\mathbb{R}^d$ which solves the Cauchy problem $L^{*}\mu = 0$ with initial condition $\mu_0 = \nu$, where $\nu$ is a probability measure on $\mathbb{R}^d$, in the following weak sense:

$$\[ \int_0^1\int_{\mathbb{R}^d} Lu(t, x)\, \mu_t(dx)\, dt=0, \quad u \in C_0^\infty((0, 1) \times \mathbb{R}^d), \] $$

and

$$\[ \lim\limits_{t \to 0} \int_{\mathbb{R}^d} \zeta(x)\, \mu_t(dx) = \int_{\mathbb{R}^d} \zeta(x)\, \nu(dx), \quad \zeta \in C_0^\infty(\mathbb{R}^d). \] $$

Such an equation is satisfied by transition probabilities of a diffusion process associated with $A$ and $b$ provided such a process exists. However, we do not assume the existence of a process and allow quite singular coefficients, in particular, $b$ may be locally unbounded or $A$ may be degenerate. An infinite-dimensional analogue is discussed as well. Main methods are $L^p$-analysis with respect to suitably chosen measures and reduction to the elliptic case (studied previously) by piecewise constant approximations in time.