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• W2015423868 abstract "The entropy solutions to the partial differential equation <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis partial-differential slash partial-differential t right-parenthesis u left-parenthesis t comma x comma y right-parenthesis plus left-parenthesis partial-differential slash partial-differential x right-parenthesis f left-parenthesis u left-parenthesis t comma x comma y right-parenthesis right-parenthesis plus left-parenthesis partial-differential slash partial-differential y right-parenthesis g left-parenthesis u left-parenthesis t comma x comma y right-parenthesis right-parenthesis equals 0 comma> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(partial /partial t)u(t,x,y) + (partial /partial x)f(u(t,x,y)) + (partial /partial y)g(u(t,x,y)) = 0,</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> with initial data constant in each quadrant of the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis x comma y right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(x,y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> plane, have been constructed and are piecewise smooth under the condition <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f left-parenthesis u right-parenthesis not-equals 0 comma g left-parenthesis u right-parenthesis not-equals 0 comma left-parenthesis f left-parenthesis u right-parenthesis slash g left-parenthesis u right-parenthesis right-parenthesis prime not-equals 0> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> <mml:mi class=MJX-variant mathvariant=normal>′<!-- ′ --></mml:mi> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>f(u) ne 0, g(u) ne 0, (f(u)/g(u))prime ne 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This problem generalizes to several space dimensions the important Riemann problem for equations in one-space dimension. Although existence and uniqueness of solutions are well known, little is known about the qualitative behavior of solutions. It is this with which we are concerned here." @default.
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• W2015423868 title "Two-dimensional Riemann problem for a single conservation law" @default.
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