We prove existence of L²-weak solutions of a quasilinear wave equation with boundary conditions. This describes the isothermal evolution of a one dimensional non-linear elastic material, attached to a fixed point on one side and subject to a force (tension) applied to the other side. The L²-valued solutions appear naturally when studying the hydrodynamic limit from a microscopic dynamics of a chain of anharmonic springs connected to a thermal bath. The proof of the existence is done using a vanishing viscosity approximation with extra Neumann boundary conditions added. In this setting we obtain a uniform a priori estimate in L², allowing us to use L² Young measures, together with the classical tools of compensated compactness. We then prove that the viscous solutions converge to weak solutions of the quasilinear wave equation strongly in L^p, for any $p\in [1,2),$ that satisfy, in a weak sense, the boundary conditions. Furthermore, these solutions satisfy, beside the local Lax entropy condition, the Clausius inequality: the change of the free energy is bounded by the work done by the boundary tension. In this sense they are the correct thermodynamic solutions, and we conjecture their uniqueness.

我们证明了带边界条件的拟线性波动方程的L ^2-弱解的存在。这描述了一维非线性弹性材料的等温演变，该材料在一侧附着到固定点，并在另一侧受到力（张力）。当从连接到热浴的非谐弹簧链的微观动力学研究流体动力学极限时，L ²值解自然而然地出现。存在性的证明是使用消失的粘度近似值加上额外的诺伊曼边界条件完成的。在这种情况下，我们在L ^2中获得统一的先验估计，从而使我们可以使用L ²年轻的措施，以及补偿紧凑性的经典工具。然后我们证明，对于任何情况，粘性解都在L ^p中强烈收敛于拟线性波动方程的弱解。$p\in [\mathrm{1个}，2），$在某种意义上满足边界条件。此外，这些解决方案除了满足局部Lax熵条件外，还满足了Clausius不等式：自由能的变化受边界张力完成的功的限制。从这个意义上讲，它们是正确的热力学解决方案，我们推测它们的独特性。